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Theorem opthreg 4308
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4290 (via the preleq 4307 step). See df-op 3412 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
Hypotheses
Ref Expression
preleq.1 𝐴 ∈ V
preleq.2 𝐵 ∈ V
preleq.3 𝐶 ∈ V
preleq.4 𝐷 ∈ V
Assertion
Ref Expression
opthreg ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem opthreg
StepHypRef Expression
1 preleq.1 . . . . 5 𝐴 ∈ V
21prid1 3504 . . . 4 𝐴 ∈ {𝐴, 𝐵}
3 preleq.3 . . . . 5 𝐶 ∈ V
43prid1 3504 . . . 4 𝐶 ∈ {𝐶, 𝐷}
5 preleq.2 . . . . . 6 𝐵 ∈ V
6 prexgOLD 3974 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
71, 5, 6mp2an 410 . . . . 5 {𝐴, 𝐵} ∈ V
8 preleq.4 . . . . . 6 𝐷 ∈ V
9 prexgOLD 3974 . . . . . 6 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → {𝐶, 𝐷} ∈ V)
103, 8, 9mp2an 410 . . . . 5 {𝐶, 𝐷} ∈ V
111, 7, 3, 10preleq 4307 . . . 4 (((𝐴 ∈ {𝐴, 𝐵} ∧ 𝐶 ∈ {𝐶, 𝐷}) ∧ {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
122, 4, 11mpanl12 420 . . 3 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
13 preq1 3475 . . . . . 6 (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
1413eqeq1d 2064 . . . . 5 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
155, 8preqr2 3568 . . . . 5 ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
1614, 15syl6bi 156 . . . 4 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
1716imdistani 427 . . 3 ((𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))
1812, 17syl 14 . 2 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶𝐵 = 𝐷))
19 preq1 3475 . . . 4 (𝐴 = 𝐶 → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
2019adantr 265 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
21 preq12 3477 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
2221preq2d 3482 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐶, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
2320, 22eqtrd 2088 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
2418, 23impbii 121 1 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1259  wcel 1409  Vcvv 2574  {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pr 3972  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-dif 2948  df-un 2950  df-sn 3409  df-pr 3410
This theorem is referenced by: (None)
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