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Theorem cardalephex 8951
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 472 . . . . 5 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ω ⊆ 𝐴)
2 cardaleph 8950 . . . . . . 7 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
32sseq2d 3666 . . . . . 6 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (ω ⊆ 𝐴 ↔ ω ⊆ (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})))
4 alephgeom 8943 . . . . . 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 6229 . . . . . 6 (𝑥 = {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} → (ℵ‘𝑥) = (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
87eqeq2d 2661 . . . . 5 (𝑥 = {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} → (𝐴 = (ℵ‘𝑥) ↔ 𝐴 = (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})))
98rspcev 3340 . . . 4 (( {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On ∧ 𝐴 = (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
106, 2, 9syl2anc 694 . . 3 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
1110ex 449 . 2 (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
12 alephcard 8931 . . . 4 (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)
13 fveq2 6229 . . . 4 (𝐴 = (ℵ‘𝑥) → (card‘𝐴) = (card‘(ℵ‘𝑥)))
14 id 22 . . . 4 (𝐴 = (ℵ‘𝑥) → 𝐴 = (ℵ‘𝑥))
1512, 13, 143eqtr4a 2711 . . 3 (𝐴 = (ℵ‘𝑥) → (card‘𝐴) = 𝐴)
1615rexlimivw 3058 . 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 383   = wceq 1523  wcel 2030  wrex 2942  {crab 2945  wss 3607   cint 4507  Oncon0 5761  cfv 5926  ωcom 7107  cardccrd 8799  cale 8800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-har 8504  df-card 8803  df-aleph 8804
This theorem is referenced by:  infenaleph  8952  isinfcard  8953  alephfp  8969  alephval3  8971  dfac12k  9007  alephval2  9432  winalim2  9556
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