Theorem isinfcard 10052

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 10025	. . 3 ⊢ ℵ Fn On
2		fvelrnb 6924	. . 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 10042	. . . . . . 7 ⊢ (𝑥 ∈ On ↔ ω ⊆ (ℵ‘𝑥))
5	4	biimpi 216	. . . . . 6 ⊢ (𝑥 ∈ On → ω ⊆ (ℵ‘𝑥))
6		sseq2 3976	. . . . . 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 3128	. . . 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 2737	. . . 4 ⊢ ((ℵ‘𝑥) = 𝐴 ↔ 𝐴 = (ℵ‘𝑥))
11	10	rexbii 3077	. . 3 ⊢ (∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
12		cardalephex 10050	. . . 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 2109  ∃wrex 3054   ⊆ wss 3917  ran crn 5642  Oncon0 6335   Fn wfn 6509  ‘cfv 6514  ωcom 7845  cardccrd 9895  ℵcale 9896

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-oi 9470  df-har 9517  df-card 9899  df-aleph 9900

This theorem is referenced by:  iscard3  10053  alephinit  10055  cardinfima  10057  alephiso  10058  alephsson  10060  alephfp  10068