Theorem cardalephex 10002

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 482	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ω ⊆ 𝐴)
2		cardaleph 10001	. . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
3	2	sseq2d 3966	. . . . . 6 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (ω ⊆ 𝐴 ↔ ω ⊆ (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})))
4		alephgeom 9994	. . . . . 6 ⊢ (∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On ↔ ω ⊆ (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
5	3, 4	bitr4di 289	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (ω ⊆ 𝐴 ↔ ∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On))
6	1, 5	mpbid 232	. . . 4 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On)
7		fveq2 6834	. . . . 5 ⊢ (𝑥 = ∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} → (ℵ‘𝑥) = (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
8	7	rspceeqv 3599	. . . 4 ⊢ ((∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On ∧ 𝐴 = (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
9	6, 2, 8	syl2anc 584	. . 3 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
10	9	ex 412	. 2 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
11		alephcard 9982	. . . 4 ⊢ (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)
12		fveq2 6834	. . . 4 ⊢ (𝐴 = (ℵ‘𝑥) → (card‘𝐴) = (card‘(ℵ‘𝑥)))
13		id 22	. . . 4 ⊢ (𝐴 = (ℵ‘𝑥) → 𝐴 = (ℵ‘𝑥))
14	11, 12, 13	3eqtr4a 2797	. . 3 ⊢ (𝐴 = (ℵ‘𝑥) → (card‘𝐴) = 𝐴)
15	14	rexlimivw 3133	. 2 ⊢ (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → (card‘𝐴) = 𝐴)
16	10, 15	impbid1 225	1 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  {crab 3399   ⊆ wss 3901  ∩ cint 4902  Oncon0 6317  ‘cfv 6492  ωcom 7808  cardccrd 9849  ℵcale 9850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-oi 9417  df-har 9464  df-card 9853  df-aleph 9854

This theorem is referenced by:  infenaleph  10003  isinfcard  10004  alephfp  10020  alephval3  10022  dfac12k  10060  alephval2  10485  winalim2  10609  minregex  43796