Theorem cardalephex 10169

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 488	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ω ⊆ 𝐴)
2		cardaleph 10168	. . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
3	2	sseq2d 3963	. . . . . 6 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (ω ⊆ 𝐴 ↔ ω ⊆ (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)})))
4		alephgeom 10161	. . . . . 6 ⊢ (∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On ↔ ω ⊆ (ℵ‘∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)}))
5	3, 4	bitr4di 292	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (ω ⊆ 𝐴 ↔ ∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On))
6	1, 5	mpbid 235	. . . 4 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∩ {𝑦 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑦)} ∈ On)
7		fveq2 6885	. . . . 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 596	. . 3 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
10	9	ex 418	. 2 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
11		alephcard 10149	. . . 4 ⊢ (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)
12		fveq2 6885	. . . 4 ⊢ (𝐴 = (ℵ‘𝑥) → (card‘𝐴) = (card‘(ℵ‘𝑥)))
13		id 23	. . . 4 ⊢ (𝐴 = (ℵ‘𝑥) → 𝐴 = (ℵ‘𝑥))
14	11, 12, 13	3eqtr4a 2822	. . 3 ⊢ (𝐴 = (ℵ‘𝑥) → (card‘𝐴) = 𝐴)
15	14	rexlimivw 3160	. 2 ⊢ (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → (card‘𝐴) = 𝐴)
16	10, 15	impbid1 228	1 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  {crab 3413   ⊆ wss 3899  ∩ cint 4907  Oncon0 6362  ‘cfv 6538  ωcom 7877  cardccrd 10016  ℵcale 10017

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-oi 9504  df-har 9551  df-card 10020  df-aleph 10021

This theorem is used by:  infenaleph  10170  isinfcard  10171  alephfp  10187  alephval3  10189  dfac12k  10226  alephval2  10657  winalim2  10781  minregex  44534