Theorem isinfcard 9918

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 9891	. . 3 ⊢ ℵ Fn On
2		fvelrnb 6867	. . 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 9908	. . . . . . 7 ⊢ (𝑥 ∈ On ↔ ω ⊆ (ℵ‘𝑥))
5	4	biimpi 215	. . . . . 6 ⊢ (𝑥 ∈ On → ω ⊆ (ℵ‘𝑥))
6		sseq2 3956	. . . . . 6 ⊢ (𝐴 = (ℵ‘𝑥) → (ω ⊆ 𝐴 ↔ ω ⊆ (ℵ‘𝑥)))
7	5, 6	syl5ibrcom 246	. . . . 5 ⊢ (𝑥 ∈ On → (𝐴 = (ℵ‘𝑥) → ω ⊆ 𝐴))
8	7	rexlimiv 3142	. . . 4 ⊢ (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ω ⊆ 𝐴)
9	8	pm4.71ri 561	. . 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 9916	. . . 4 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
13	12	pm5.32i 575	. . 3 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ (ω ⊆ 𝐴 ∧ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
14	9, 11, 13	3bitr4i 302	. 2 ⊢ (∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴 ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
15	3, 14	bitr2i 275	1 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∃wrex 3071   ⊆ wss 3896  ran crn 5606  Oncon0 6286   Fn wfn 6458  ‘cfv 6463  ωcom 7755  cardccrd 9761  ℵcale 9762

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626  ax-inf2 9467

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-int 4891  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-tr 5203  df-id 5505  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5560  df-se 5561  df-we 5562  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-pred 6222  df-ord 6289  df-on 6290  df-lim 6291  df-suc 6292  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-isom 6472  df-riota 7270  df-ov 7316  df-om 7756  df-2nd 7875  df-frecs 8142  df-wrecs 8173  df-recs 8247  df-rdg 8286  df-1o 8342  df-er 8544  df-en 8780  df-dom 8781  df-sdom 8782  df-fin 8783  df-oi 9337  df-har 9384  df-card 9765  df-aleph 9766

This theorem is referenced by:  iscard3  9919  alephinit  9921  cardinfima  9923  alephiso  9924  alephsson  9926  alephfp  9934