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Theorem alephiso2 42611
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 10095 . 2 β„΅ Isom E , E (On, {π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)})
2 iscard4 42586 . . . . . . . 8 ((cardβ€˜π‘₯) = π‘₯ ↔ π‘₯ ∈ ran card)
32anbi1ci 624 . . . . . . 7 ((Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯) ↔ (π‘₯ ∈ ran card ∧ Ο‰ βŠ† π‘₯))
43abbii 2800 . . . . . 6 {π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)} = {π‘₯ ∣ (π‘₯ ∈ ran card ∧ Ο‰ βŠ† π‘₯)}
5 df-rab 3431 . . . . . 6 {π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯} = {π‘₯ ∣ (π‘₯ ∈ ran card ∧ Ο‰ βŠ† π‘₯)}
64, 5eqtr4i 2761 . . . . 5 {π‘₯ ∣ (Ο‰ βŠ† π‘₯ ∧ (cardβ€˜π‘₯) = π‘₯)} = {π‘₯ ∈ ran card ∣ Ο‰ βŠ† π‘₯}
7 f1oeq3 6822 . . . . 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 10066 . . . . . . . . 9 (β„΅β€˜π‘§) ∈ On
10 epelg 5580 . . . . . . . . 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 10073 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑧 ∈ On) β†’ (𝑦 ∈ 𝑧 ↔ (β„΅β€˜π‘¦) ∈ (β„΅β€˜π‘§)))
13 alephord 10072 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑧 ∈ On) β†’ (𝑦 ∈ 𝑧 ↔ (β„΅β€˜π‘¦) β‰Ί (β„΅β€˜π‘§)))
1411, 12, 133bitr2d 306 . . . . . . 7 ((𝑦 ∈ On ∧ 𝑧 ∈ On) β†’ ((β„΅β€˜π‘¦) E (β„΅β€˜π‘§) ↔ (β„΅β€˜π‘¦) β‰Ί (β„΅β€˜π‘§)))
1514bibi2d 341 . . . . . 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 3089 . . . 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 625 . . 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 6551 . . 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 6551 . . 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 302 . 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 394   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆ€wral 3059  {crab 3430   βŠ† wss 3947   class class class wbr 5147   E cep 5578  ran crn 5676  Oncon0 6363  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542   Isom wiso 6543  Ο‰com 7857   β‰Ί csdm 8940  cardccrd 9932  β„΅cale 9933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-har 9554  df-card 9936  df-aleph 9937
This theorem is referenced by:  alephiso3  42612
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