Theorem isinfcard 10132

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 10105	. . 3 ⊢ ℵ Fn On
2		fvelrnb 6969	. . 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 10122	. . . . . . 7 ⊢ (𝑥 ∈ On ↔ ω ⊆ (ℵ‘𝑥))
5	4	biimpi 216	. . . . . 6 ⊢ (𝑥 ∈ On → ω ⊆ (ℵ‘𝑥))
6		sseq2 4010	. . . . . 6 ⊢ (𝐴 = (ℵ‘𝑥) → (ω ⊆ 𝐴 ↔ ω ⊆ (ℵ‘𝑥)))
7	5, 6	syl5ibrcom 247	. . . . 5 ⊢ (𝑥 ∈ On → (𝐴 = (ℵ‘𝑥) → ω ⊆ 𝐴))
8	7	rexlimiv 3148	. . . 4 ⊢ (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ω ⊆ 𝐴)
9	8	pm4.71ri 560	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) ↔ (ω ⊆ 𝐴 ∧ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
10		eqcom 2744	. . . 4 ⊢ ((ℵ‘𝑥) = 𝐴 ↔ 𝐴 = (ℵ‘𝑥))
11	10	rexbii 3094	. . 3 ⊢ (∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
12		cardalephex 10130	. . . 4 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
13	12	pm5.32i 574	. . 3 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ (ω ⊆ 𝐴 ∧ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
14	9, 11, 13	3bitr4i 303	. 2 ⊢ (∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴 ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
15	3, 14	bitr2i 276	1 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ⊆ wss 3951  ran crn 5686  Oncon0 6384   Fn wfn 6556  ‘cfv 6561  ωcom 7887  cardccrd 9975  ℵcale 9976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-har 9597  df-card 9979  df-aleph 9980

This theorem is referenced by:  iscard3  10133  alephinit  10135  cardinfima  10137  alephiso  10138  alephsson  10140  alephfp  10148