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Theorem cardalephex 8858
Description: Every transfinite cardinal is an aleph and vice-versa. Theorem 8A(b) of [Enderton] p. 213 and its converse. (Contributed by NM, 5-Nov-2003.)
Assertion
Ref Expression
cardalephex (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem cardalephex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpl 473 . . . . 5 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ω ⊆ 𝐴)
2 cardaleph 8857 . . . . . . 7 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
32sseq2d 3617 . . . . . 6 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (ω ⊆ 𝐴 ↔ ω ⊆ (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})))
4 alephgeom 8850 . . . . . 6 ( {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On ↔ ω ⊆ (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
53, 4syl6bbr 278 . . . . 5 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (ω ⊆ 𝐴 {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On))
61, 5mpbid 222 . . . 4 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On)
7 fveq2 6150 . . . . . 6 (𝑥 = {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} → (ℵ‘𝑥) = (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
87eqeq2d 2636 . . . . 5 (𝑥 = {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} → (𝐴 = (ℵ‘𝑥) ↔ 𝐴 = (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})))
98rspcev 3300 . . . 4 (( {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On ∧ 𝐴 = (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
106, 2, 9syl2anc 692 . . 3 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
1110ex 450 . 2 (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
12 alephcard 8838 . . . 4 (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)
13 fveq2 6150 . . . 4 (𝐴 = (ℵ‘𝑥) → (card‘𝐴) = (card‘(ℵ‘𝑥)))
14 id 22 . . . 4 (𝐴 = (ℵ‘𝑥) → 𝐴 = (ℵ‘𝑥))
1512, 13, 143eqtr4a 2686 . . 3 (𝐴 = (ℵ‘𝑥) → (card‘𝐴) = 𝐴)
1615rexlimivw 3027 . 2 (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → (card‘𝐴) = 𝐴)
1711, 16impbid1 215 1 (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wrex 2913  {crab 2916  wss 3560   cint 4445  Oncon0 5685  cfv 5850  ωcom 7013  cardccrd 8706  cale 8707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-oi 8360  df-har 8408  df-card 8710  df-aleph 8711
This theorem is referenced by:  infenaleph  8859  isinfcard  8860  alephfp  8876  alephval3  8878  dfac12k  8914  alephval2  9339  winalim2  9463
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