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Theorem cardalephex 9199
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 475 . . . . 5 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ω ⊆ 𝐴)
2 cardaleph 9198 . . . . . . 7 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
32sseq2d 3829 . . . . . 6 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (ω ⊆ 𝐴 ↔ ω ⊆ (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})))
4 alephgeom 9191 . . . . . 6 ( {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On ↔ ω ⊆ (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
53, 4syl6bbr 281 . . . . 5 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (ω ⊆ 𝐴 {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On))
61, 5mpbid 224 . . . 4 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On)
7 fveq2 6411 . . . . 5 (𝑥 = {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} → (ℵ‘𝑥) = (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
87rspceeqv 3515 . . . 4 (( {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On ∧ 𝐴 = (ℵ‘ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
96, 2, 8syl2anc 580 . . 3 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
109ex 402 . 2 (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
11 alephcard 9179 . . . 4 (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)
12 fveq2 6411 . . . 4 (𝐴 = (ℵ‘𝑥) → (card‘𝐴) = (card‘(ℵ‘𝑥)))
13 id 22 . . . 4 (𝐴 = (ℵ‘𝑥) → 𝐴 = (ℵ‘𝑥))
1411, 12, 133eqtr4a 2859 . . 3 (𝐴 = (ℵ‘𝑥) → (card‘𝐴) = 𝐴)
1514rexlimivw 3210 . 2 (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → (card‘𝐴) = 𝐴)
1610, 15impbid1 217 1 (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wrex 3090  {crab 3093  wss 3769   cint 4667  Oncon0 5941  cfv 6101  ωcom 7299  cardccrd 9047  cale 9048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-inf2 8788
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-oi 8657  df-har 8705  df-card 9051  df-aleph 9052
This theorem is referenced by:  infenaleph  9200  isinfcard  9201  alephfp  9217  alephval3  9219  dfac12k  9257  alephval2  9682  winalim2  9806
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