Theorem cardalephex 8913

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.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ω ⊆ 𝐴)
2		cardaleph 8912	. . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
3	2	sseq2d 3633	. . . . . 6 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (ω ⊆ 𝐴 ↔ ω ⊆ (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})))
4		alephgeom 8905	. . . . . 6 ⊢ (∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On ↔ ω ⊆ (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
5	3, 4	syl6bbr 278	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (ω ⊆ 𝐴 ↔ ∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On))
6	1, 5	mpbid 222	. . . 4 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On)
7		fveq2 6191	. . . . . 6 ⊢ (𝑥 = ∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} → (ℵ‘𝑥) = (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
8	7	eqeq2d 2632	. . . . 5 ⊢ (𝑥 = ∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} → (𝐴 = (ℵ‘𝑥) ↔ 𝐴 = (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})))
9	8	rspcev 3309	. . . 4 ⊢ ((∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On ∧ 𝐴 = (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
10	6, 2, 9	syl2anc 693	. . 3 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
11	10	ex 450	. 2 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
12		alephcard 8893	. . . 4 ⊢ (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)
13		fveq2 6191	. . . 4 ⊢ (𝐴 = (ℵ‘𝑥) → (card‘𝐴) = (card‘(ℵ‘𝑥)))
14		id 22	. . . 4 ⊢ (𝐴 = (ℵ‘𝑥) → 𝐴 = (ℵ‘𝑥))
15	12, 13, 14	3eqtr4a 2682	. . 3 ⊢ (𝐴 = (ℵ‘𝑥) → (card‘𝐴) = 𝐴)
16	15	rexlimivw 3029	. 2 ⊢ (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → (card‘𝐴) = 𝐴)
17	11, 16	impbid1 215	1 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  {crab 2916   ⊆ wss 3574  ∩ cint 4475  Oncon0 5723  ‘cfv 5888  ωcom 7065  cardccrd 8761  ℵcale 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766

This theorem is referenced by:  infenaleph  8914  isinfcard  8915  alephfp  8931  alephval3  8933  dfac12k  8969  alephval2  9394  winalim2  9518