Theorem isinfcard 9248

Description: Two ways to express the property of being a transfinite cardinal. (Contributed by NM, 9-Nov-2003.)

Assertion

Ref	Expression
isinfcard	⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)

Proof of Theorem isinfcard

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephfnon 9221	. . 3 ⊢ ℵ Fn On
2		fvelrnb 6503	. . 3 ⊢ (ℵ Fn On → (𝐴 ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐴 ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴)
4		alephgeom 9238	. . . . . . 7 ⊢ (𝑥 ∈ On ↔ ω ⊆ (ℵ‘𝑥))
5	4	biimpi 208	. . . . . 6 ⊢ (𝑥 ∈ On → ω ⊆ (ℵ‘𝑥))
6		sseq2 3846	. . . . . 6 ⊢ (𝐴 = (ℵ‘𝑥) → (ω ⊆ 𝐴 ↔ ω ⊆ (ℵ‘𝑥)))
7	5, 6	syl5ibrcom 239	. . . . 5 ⊢ (𝑥 ∈ On → (𝐴 = (ℵ‘𝑥) → ω ⊆ 𝐴))
8	7	rexlimiv 3209	. . . 4 ⊢ (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ω ⊆ 𝐴)
9	8	pm4.71ri 556	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) ↔ (ω ⊆ 𝐴 ∧ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
10		eqcom 2785	. . . 4 ⊢ ((ℵ‘𝑥) = 𝐴 ↔ 𝐴 = (ℵ‘𝑥))
11	10	rexbii 3224	. . 3 ⊢ (∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
12		cardalephex 9246	. . . 4 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
13	12	pm5.32i 570	. . 3 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ (ω ⊆ 𝐴 ∧ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
14	9, 11, 13	3bitr4i 295	. 2 ⊢ (∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴 ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
15	3, 14	bitr2i 268	1 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)

Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∃wrex 3091   ⊆ wss 3792  ran crn 5356  Oncon0 5976   Fn wfn 6130  ‘cfv 6135  ωcom 7343  cardccrd 9094  ℵcale 9095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-oi 8704  df-har 8752  df-card 9098  df-aleph 9099

This theorem is referenced by:  iscard3  9249  alephinit  9251  cardinfima  9253  alephiso  9254  alephsson  9256  alephfp  9264