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Theorem alephiso2 41837
Description: β„΅ is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.)
Assertion
Ref Expression
alephiso2 β„΅ Isom E , β‰Ί (On, {π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯})

Proof of Theorem alephiso2
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alephiso 10035 . 2 β„΅ Isom E , E (On, {π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)})
2 iscard4 41812 . . . . . . . 8 ((cardβ€˜π‘₯) = π‘₯ ↔ π‘₯ ∈ ran card)
32anbi1ci 627 . . . . . . 7 ((Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯) ↔ (π‘₯ ∈ ran card ∧ Ο‰ βŠ† π‘₯))
43abbii 2807 . . . . . 6 {π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)} = {π‘₯ ∣ (π‘₯ ∈ ran card ∧ Ο‰ βŠ† π‘₯)}
5 df-rab 3409 . . . . . 6 {π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯} = {π‘₯ ∣ (π‘₯ ∈ ran card ∧ Ο‰ βŠ† π‘₯)}
64, 5eqtr4i 2768 . . . . 5 {π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)} = {π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯}
7 f1oeq3 6775 . . . . 5 ({π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)} = {π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯} β†’ (β„΅:On–1-1-ontoβ†’{π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)} ↔ β„΅:On–1-1-ontoβ†’{π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯}))
86, 7ax-mp 5 . . . 4 (β„΅:On–1-1-ontoβ†’{π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)} ↔ β„΅:On–1-1-ontoβ†’{π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯})
9 alephon 10006 . . . . . . . . 9 (β„΅β€˜π‘§) ∈ On
10 epelg 5539 . . . . . . . . 9 ((β„΅β€˜π‘§) ∈ On β†’ ((β„΅β€˜π‘¦) E (β„΅β€˜π‘§) ↔ (β„΅β€˜π‘¦) ∈ (β„΅β€˜π‘§)))
119, 10mp1i 13 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑧 ∈ On) β†’ ((β„΅β€˜π‘¦) E (β„΅β€˜π‘§) ↔ (β„΅β€˜π‘¦) ∈ (β„΅β€˜π‘§)))
12 alephord2 10013 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑧 ∈ On) β†’ (𝑦 ∈ 𝑧 ↔ (β„΅β€˜π‘¦) ∈ (β„΅β€˜π‘§)))
13 alephord 10012 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑧 ∈ On) β†’ (𝑦 ∈ 𝑧 ↔ (β„΅β€˜π‘¦) β‰Ί (β„΅β€˜π‘§)))
1411, 12, 133bitr2d 307 . . . . . . 7 ((𝑦 ∈ On ∧ 𝑧 ∈ On) β†’ ((β„΅β€˜π‘¦) E (β„΅β€˜π‘§) ↔ (β„΅β€˜π‘¦) β‰Ί (β„΅β€˜π‘§)))
1514bibi2d 343 . . . . . 6 ((𝑦 ∈ On ∧ 𝑧 ∈ On) β†’ ((𝑦 E 𝑧 ↔ (β„΅β€˜π‘¦) E (β„΅β€˜π‘§)) ↔ (𝑦 E 𝑧 ↔ (β„΅β€˜π‘¦) β‰Ί (β„΅β€˜π‘§))))
1615ralbidva 3173 . . . . 5 (𝑦 ∈ On β†’ (βˆ€π‘§ ∈ On (𝑦 E 𝑧 ↔ (β„΅β€˜π‘¦) E (β„΅β€˜π‘§)) ↔ βˆ€π‘§ ∈ On (𝑦 E 𝑧 ↔ (β„΅β€˜π‘¦) β‰Ί (β„΅β€˜π‘§))))
1716ralbiia 3095 . . . 4 (βˆ€π‘¦ ∈ On βˆ€π‘§ ∈ On (𝑦 E 𝑧 ↔ (β„΅β€˜π‘¦) E (β„΅β€˜π‘§)) ↔ βˆ€π‘¦ ∈ On βˆ€π‘§ ∈ On (𝑦 E 𝑧 ↔ (β„΅β€˜π‘¦) β‰Ί (β„΅β€˜π‘§)))
188, 17anbi12i 628 . . 3 ((β„΅:On–1-1-ontoβ†’{π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)} ∧ βˆ€π‘¦ ∈ On βˆ€π‘§ ∈ On (𝑦 E 𝑧 ↔ (β„΅β€˜π‘¦) E (β„΅β€˜π‘§))) ↔ (β„΅:On–1-1-ontoβ†’{π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯} ∧ βˆ€π‘¦ ∈ On βˆ€π‘§ ∈ On (𝑦 E 𝑧 ↔ (β„΅β€˜π‘¦) β‰Ί (β„΅β€˜π‘§))))
19 df-isom 6506 . . 3 (β„΅ Isom E , E (On, {π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)}) ↔ (β„΅:On–1-1-ontoβ†’{π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)} ∧ βˆ€π‘¦ ∈ On βˆ€π‘§ ∈ On (𝑦 E 𝑧 ↔ (β„΅β€˜π‘¦) E (β„΅β€˜π‘§))))
20 df-isom 6506 . . 3 (β„΅ Isom E , β‰Ί (On, {π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯}) ↔ (β„΅:On–1-1-ontoβ†’{π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯} ∧ βˆ€π‘¦ ∈ On βˆ€π‘§ ∈ On (𝑦 E 𝑧 ↔ (β„΅β€˜π‘¦) β‰Ί (β„΅β€˜π‘§))))
2118, 19, 203bitr4i 303 . 2 (β„΅ Isom E , E (On, {π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)}) ↔ β„΅ Isom E , β‰Ί (On, {π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯}))
221, 21mpbi 229 1 β„΅ Isom E , β‰Ί (On, {π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯})
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2714  βˆ€wral 3065  {crab 3408   βŠ† wss 3911   class class class wbr 5106   E cep 5537  ran crn 5635  Oncon0 6318  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497   Isom wiso 6498  Ο‰com 7803   β‰Ί csdm 8883  cardccrd 9872  β„΅cale 9873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9578
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8649  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-oi 9447  df-har 9494  df-card 9876  df-aleph 9877
This theorem is referenced by:  alephiso3  41838
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