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Theorem List for Intuitionistic Logic Explorer - 5501-5600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisoid 5501 Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴)
 
Theoremisocnv 5502 Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
 
Theoremisocnv2 5503 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵))
 
Theoremisores2 5504 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
 
Theoremisores1 5505 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
 
Theoremisores3 5506 Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴𝑋 = (𝐻𝐾)) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))
 
Theoremisotr 5507 Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑆, 𝑇 (𝐵, 𝐶)) → (𝐺𝐻) Isom 𝑅, 𝑇 (𝐴, 𝐶))
 
Theoremisoini 5508 Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝐷}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})))
 
Theoremisoini2 5509 Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)
𝐶 = (𝐴 ∩ (𝑅 “ {𝑋}))    &   𝐷 = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)}))       ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))
 
Theoremisoselem 5510* Lemma for isose 5511. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑 → (𝐻𝑥) ∈ V)       (𝜑 → (𝑅 Se 𝐴𝑆 Se 𝐵))
 
Theoremisose 5511 An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴𝑆 Se 𝐵))
 
Theoremisopolem 5512 Lemma for isopo 5513. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))
 
Theoremisopo 5513 An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Po 𝐴𝑆 Po 𝐵))
 
Theoremisosolem 5514 Lemma for isoso 5515. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
 
Theoremisoso 5515 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴𝑆 Or 𝐵))
 
Theoremf1oiso 5516* Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)
((𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
 
Theoremf1oiso2 5517* Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. (Contributed by Mario Carneiro, 9-Mar-2013.)
𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦))}       (𝐻:𝐴1-1-onto𝐵𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
 
2.6.9  Restricted iota (description binder)
 
Syntaxcrio 5518 Extend class notation with restricted description binder.
class (𝑥𝐴 𝜑)
 
Definitiondf-riota 5519 Define restricted description binder. In case there is no unique 𝑥 such that (𝑥𝐴𝜑) holds, it evaluates to the empty set. See also comments for df-iota 4917. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)
(𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
 
Theoremriotaeqdv 5520* Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜓))
 
Theoremriotabidv 5521* Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
 
Theoremriotaeqbidv 5522* Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜒))
 
Theoremriotaexg 5523* Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
(𝐴𝑉 → (𝑥𝐴 𝜓) ∈ V)
 
Theoremriotav 5524 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
(𝑥 ∈ V 𝜑) = (℩𝑥𝜑)
 
Theoremriotauni 5525 Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
(∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})
 
Theoremnfriota1 5526* The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥(𝑥𝐴 𝜑)
 
Theoremnfriotadxy 5527* Deduction version of nfriota 5528. (Contributed by Jim Kingdon, 12-Jan-2019.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑𝑥𝐴)       (𝜑𝑥(𝑦𝐴 𝜓))
 
Theoremnfriota 5528* A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)
𝑥𝜑    &   𝑥𝐴       𝑥(𝑦𝐴 𝜑)
 
Theoremcbvriota 5529* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
 
Theoremcbvriotav 5530* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
 
Theoremcsbriotag 5531* Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
(𝐴𝑉𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
 
Theoremriotacl2 5532 Membership law for "the unique element in 𝐴 such that 𝜑."

(Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

(∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
 
Theoremriotacl 5533* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
(∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
 
Theoremriotasbc 5534 Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
 
Theoremriotabidva 5535* Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2598 analog.) (Contributed by NM, 17-Jan-2012.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
 
Theoremriotabiia 5536 Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2596 analog.) (Contributed by NM, 16-Jan-2012.)
(𝑥𝐴 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑥𝐴 𝜓)
 
Theoremriota1 5537* Property of restricted iota. Compare iota1 4931. (Contributed by Mario Carneiro, 15-Oct-2016.)
(∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (𝑥𝐴 𝜑) = 𝑥))
 
Theoremriota1a 5538 Property of iota. (Contributed by NM, 23-Aug-2011.)
((𝑥𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
 
Theoremriota2df 5539* A deduction version of riota2f 5540. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝜑    &   (𝜑𝑥𝐵)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑𝐵𝐴)    &   ((𝜑𝑥 = 𝐵) → (𝜓𝜒))       ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (𝑥𝐴 𝜓) = 𝐵))
 
Theoremriota2f 5540* This theorem shows a condition that allows us to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐵    &   𝑥𝜓    &   (𝑥 = 𝐵 → (𝜑𝜓))       ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
 
Theoremriota2 5541* This theorem shows a condition that allows us to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.)
(𝑥 = 𝐵 → (𝜑𝜓))       ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
 
Theoremriotaprop 5542* Properties of a restricted definite description operator. Todo (df-riota 5519 update): can some uses of riota2f 5540 be shortened with this? (Contributed by NM, 23-Nov-2013.)
𝑥𝜓    &   𝐵 = (𝑥𝐴 𝜑)    &   (𝑥 = 𝐵 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 → (𝐵𝐴𝜓))
 
Theoremriota5f 5543* A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐵)    &   (𝜑𝐵𝐴)    &   ((𝜑𝑥𝐴) → (𝜓𝑥 = 𝐵))       (𝜑 → (𝑥𝐴 𝜓) = 𝐵)
 
Theoremriota5 5544* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)
(𝜑𝐵𝐴)    &   ((𝜑𝑥𝐴) → (𝜓𝑥 = 𝐵))       (𝜑 → (𝑥𝐴 𝜓) = 𝐵)
 
Theoremriotass2 5545* Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
(((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜓))
 
Theoremriotass 5546* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
 
Theoremmoriotass 5547* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
 
Theoremsnriota 5548 A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
(∃!𝑥𝐴 𝜑 → {𝑥𝐴𝜑} = {(𝑥𝐴 𝜑)})
 
Theoremeusvobj2 5549* Specify the same property in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
𝐵 ∈ V       (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
 
Theoremeusvobj1 5550* Specify the same object in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
𝐵 ∈ V       (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (℩𝑥𝑦𝐴 𝑥 = 𝐵) = (℩𝑥𝑦𝐴 𝑥 = 𝐵))
 
Theoremf1ofveu 5551* There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)
((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → ∃!𝑥𝐴 (𝐹𝑥) = 𝐶)
 
Theoremf1ocnvfv3 5552* Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝑥𝐴 (𝐹𝑥) = 𝐶))
 
Theoremriotaund 5553* Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.)
(¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
 
Theoremacexmidlema 5554* Lemma for acexmid 5562. (Contributed by Jim Kingdon, 6-Aug-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       ({∅} ∈ 𝐴𝜑)
 
Theoremacexmidlemb 5555* Lemma for acexmid 5562. (Contributed by Jim Kingdon, 6-Aug-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       (∅ ∈ 𝐵𝜑)
 
Theoremacexmidlemph 5556* Lemma for acexmid 5562. (Contributed by Jim Kingdon, 6-Aug-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       (𝜑𝐴 = 𝐵)
 
Theoremacexmidlemab 5557* Lemma for acexmid 5562. (Contributed by Jim Kingdon, 6-Aug-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       (((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}) → ¬ 𝜑)
 
Theoremacexmidlemcase 5558* Lemma for acexmid 5562. Here we divide the proof into cases (based on the disjunction implicit in an unordered pair, not the sort of case elimination which relies on excluded middle).

The cases are (1) the choice function evaluated at 𝐴 equals {∅}, (2) the choice function evaluated at 𝐵 equals , and (3) the choice function evaluated at 𝐴 equals and the choice function evaluated at 𝐵 equals {∅}.

Because of the way we represent the choice function 𝑦, the choice function evaluated at 𝐴 is (𝑣𝐴𝑢𝑦(𝐴𝑢𝑣𝑢)) and the choice function evaluated at 𝐵 is (𝑣𝐵𝑢𝑦(𝐵𝑢𝑣𝑢)). Other than the difference in notation these work just as (𝑦𝐴) and (𝑦𝐵) would if 𝑦 were a function as defined by df-fun 4954.

Although it isn't exactly about the division into cases, it is also convenient for this lemma to also include the step that if the choice function evaluated at 𝐴 equals {∅}, then {∅} ∈ 𝐴 and likewise for 𝐵.

(Contributed by Jim Kingdon, 7-Aug-2019.)

𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})))
 
Theoremacexmidlem1 5559* Lemma for acexmid 5562. List the cases identified in acexmidlemcase 5558 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (𝜑 ∨ ¬ 𝜑))
 
Theoremacexmidlem2 5560* Lemma for acexmid 5562. This builds on acexmidlem1 5559 by noting that every element of 𝐶 is inhabited.

(Note that 𝑦 is not quite a function in the df-fun 4954 sense because it uses ordered pairs as described in opthreg 4327 rather than df-op 3425).

The set 𝐴 is also found in onsucelsucexmidlem 4300.

(Contributed by Jim Kingdon, 5-Aug-2019.)

𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       (∀𝑧𝐶𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (𝜑 ∨ ¬ 𝜑))
 
Theoremacexmidlemv 5561* Lemma for acexmid 5562.

This is acexmid 5562 with additional distinct variable constraints, most notably between 𝜑 and 𝑥.

(Contributed by Jim Kingdon, 6-Aug-2019.)

𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)       (𝜑 ∨ ¬ 𝜑)
 
Theoremacexmid 5562* The axiom of choice implies excluded middle. Theorem 1.3 in [Bauer] p. 483.

The statement of the axiom of choice given here is ac2 in the Metamath Proof Explorer (version of 3-Aug-2019). In particular, note that the choice function 𝑦 provides a value when 𝑧 is inhabited (as opposed to non-empty as in some statements of the axiom of choice).

Essentially the same proof can also be found at "The axiom of choice implies instances of EM", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill theorem, after Radu Diaconescu who discovered it in 1975 in the framework of topos theory and N. D. Goodman and John Myhill in 1978 in the framework of set theory (although it already appeared as an exercise in Errett Bishop's book Foundations of Constructive Analysis from 1967).

(Contributed by Jim Kingdon, 4-Aug-2019.)

𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)       (𝜑 ∨ ¬ 𝜑)
 
2.6.10  Operations
 
Syntaxco 5563 Extend class notation to include the value of an operation 𝐹 (such as + ) for two arguments 𝐴 and 𝐵. Note that the syntax is simply three class symbols in a row surrounded by parentheses. Since operation values are the only possible class expressions consisting of three class expressions in a row surrounded by parentheses, the syntax is unambiguous.
class (𝐴𝐹𝐵)
 
Syntaxcoprab 5564 Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Syntaxcmpt2 5565 Extend the definition of a class to include maps-to notation for defining an operation via a rule.
class (𝑥𝐴, 𝑦𝐵𝐶)
 
Definitiondf-ov 5566 Define the value of an operation. Definition of operation value in [Enderton] p. 79. Note that the syntax is simply three class expressions in a row bracketed by parentheses. There are no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation 𝐹 and its arguments 𝐴 and 𝐵- will be useful for proving meaningful theorems. For example, if class 𝐹 is the operation + and arguments 𝐴 and 𝐵 are 3 and 2 , the expression ( 3 + 2 ) can be proved to equal 5 . This definition is well-defined, although not very meaningful, when classes 𝐴 and/or 𝐵 are proper classes (i.e. are not sets); see ovprc1 5592 and ovprc2 5593. On the other hand, we often find uses for this definition when 𝐹 is a proper class. 𝐹 is normally equal to a class of nested ordered pairs of the form defined by df-oprab 5567. (Contributed by NM, 28-Feb-1995.)
(𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
 
Definitiondf-oprab 5567* Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally 𝑥, 𝑦, and 𝑧 are distinct, although the definition doesn't strictly require it. See df-ov 5566 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of the most common operation class builder is given by ovmpt2 5687. (Contributed by NM, 12-Mar-1995.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
 
Definitiondf-mpt2 5568* Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from 𝑥, 𝑦 (in 𝐴 × 𝐵) to 𝐵(𝑥, 𝑦)." An extension of df-mpt 3861 for two arguments. (Contributed by NM, 17-Feb-2008.)
(𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
 
Theoremoveq 5569 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
 
Theoremoveq1 5570 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐴 = 𝐵 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
 
Theoremoveq2 5571 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐴 = 𝐵 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
 
Theoremoveq12 5572 Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveq1i 5573 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
𝐴 = 𝐵       (𝐴𝐹𝐶) = (𝐵𝐹𝐶)
 
Theoremoveq2i 5574 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
𝐴 = 𝐵       (𝐶𝐹𝐴) = (𝐶𝐹𝐵)
 
Theoremoveq12i 5575 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐹𝐶) = (𝐵𝐹𝐷)
 
Theoremoveqi 5576 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
𝐴 = 𝐵       (𝐶𝐴𝐷) = (𝐶𝐵𝐷)
 
Theoremoveq123i 5577 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
𝐴 = 𝐶    &   𝐵 = 𝐷    &   𝐹 = 𝐺       (𝐴𝐹𝐵) = (𝐶𝐺𝐷)
 
Theoremoveq1d 5578 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
 
Theoremoveq2d 5579 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
 
Theoremoveqd 5580 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷))
 
Theoremoveq12d 5581 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveqan12d 5582 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveqan12rd 5583 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveq123d 5584 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷))
 
Theoremnfovd 5585 Deduction version of bound-variable hypothesis builder nfov 5586. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐹)    &   (𝜑𝑥𝐵)       (𝜑𝑥(𝐴𝐹𝐵))
 
Theoremnfov 5586 Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐵       𝑥(𝐴𝐹𝐵)
 
Theoremoprabidlem 5587* Slight elaboration of exdistrfor 1723. A lemma for oprabid 5588. (Contributed by Jim Kingdon, 15-Jan-2019.)
(∃𝑥𝑦(𝑥 = 𝑧𝜓) → ∃𝑥(𝑥 = 𝑧 ∧ ∃𝑦𝜓))
 
Theoremoprabid 5588 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable constraint between 𝑥, 𝑦, and 𝑧, we use ax-bndl 1440 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)
(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
 
Theoremfnovex 5589 The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.)
((𝐹 Fn (𝐶 × 𝐷) ∧ 𝐴𝐶𝐵𝐷) → (𝐴𝐹𝐵) ∈ V)
 
Theoremovexg 5590 Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.)
((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐴𝐹𝐵) ∈ V)
 
Theoremovprc 5591 The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.)
Rel dom 𝐹       (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
 
Theoremovprc1 5592 The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.)
Rel dom 𝐹       𝐴 ∈ V → (𝐴𝐹𝐵) = ∅)
 
Theoremovprc2 5593 The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
Rel dom 𝐹       𝐵 ∈ V → (𝐴𝐹𝐵) = ∅)
 
Theoremcsbov123g 5594 Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
(𝐴𝐷𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
 
Theoremcsbov12g 5595* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
 
Theoremcsbov1g 5596* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
 
Theoremcsbov2g 5597* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
 
Theoremrspceov 5598* A frequently used special case of rspc2ev 2723 for operation values. (Contributed by NM, 21-Mar-2007.)
((𝐶𝐴𝐷𝐵𝑆 = (𝐶𝐹𝐷)) → ∃𝑥𝐴𝑦𝐵 𝑆 = (𝑥𝐹𝑦))
 
Theoremfnotovb 5599 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5267. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
 
Theoremopabbrex 5600* A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝𝜃))    &   ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)       ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)} ∈ V)
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