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Theorem isinfcard 8866
 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.
StepHypRef Expression
1 alephfnon 8839 . . 3 ℵ Fn On
2 fvelrnb 6205 . . 3 (ℵ Fn On → (𝐴 ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴))
31, 2ax-mp 5 . 2 (𝐴 ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴)
4 alephgeom 8856 . . . . . . 7 (𝑥 ∈ On ↔ ω ⊆ (ℵ‘𝑥))
54biimpi 206 . . . . . 6 (𝑥 ∈ On → ω ⊆ (ℵ‘𝑥))
6 sseq2 3611 . . . . . 6 (𝐴 = (ℵ‘𝑥) → (ω ⊆ 𝐴 ↔ ω ⊆ (ℵ‘𝑥)))
75, 6syl5ibrcom 237 . . . . 5 (𝑥 ∈ On → (𝐴 = (ℵ‘𝑥) → ω ⊆ 𝐴))
87rexlimiv 3021 . . . 4 (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ω ⊆ 𝐴)
98pm4.71ri 664 . . 3 (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) ↔ (ω ⊆ 𝐴 ∧ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
10 eqcom 2628 . . . 4 ((ℵ‘𝑥) = 𝐴𝐴 = (ℵ‘𝑥))
1110rexbii 3035 . . 3 (∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
12 cardalephex 8864 . . . 4 (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
1312pm5.32i 668 . . 3 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ (ω ⊆ 𝐴 ∧ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
149, 11, 133bitr4i 292 . 2 (∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴 ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
153, 14bitr2i 265 1 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∃wrex 2908   ⊆ wss 3559  ran crn 5080  Oncon0 5687   Fn wfn 5847  ‘cfv 5852  ωcom 7019  cardccrd 8712  ℵcale 8713 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-oi 8366  df-har 8414  df-card 8716  df-aleph 8717 This theorem is referenced by:  iscard3  8867  alephinit  8869  cardinfima  8871  alephiso  8872  alephsson  8874  alephfp  8882
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