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Theorem isinfcard 9503
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 9476 . . 3 ℵ Fn On
2 fvelrnb 6701 . . 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 9493 . . . . . . 7 (𝑥 ∈ On ↔ ω ⊆ (ℵ‘𝑥))
54biimpi 219 . . . . . 6 (𝑥 ∈ On → ω ⊆ (ℵ‘𝑥))
6 sseq2 3941 . . . . . 6 (𝐴 = (ℵ‘𝑥) → (ω ⊆ 𝐴 ↔ ω ⊆ (ℵ‘𝑥)))
75, 6syl5ibrcom 250 . . . . 5 (𝑥 ∈ On → (𝐴 = (ℵ‘𝑥) → ω ⊆ 𝐴))
87rexlimiv 3239 . . . 4 (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ω ⊆ 𝐴)
98pm4.71ri 564 . . 3 (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) ↔ (ω ⊆ 𝐴 ∧ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
10 eqcom 2805 . . . 4 ((ℵ‘𝑥) = 𝐴𝐴 = (ℵ‘𝑥))
1110rexbii 3210 . . 3 (∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
12 cardalephex 9501 . . . 4 (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
1312pm5.32i 578 . . 3 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ (ω ⊆ 𝐴 ∧ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
149, 11, 133bitr4i 306 . 2 (∃𝑥 ∈ On (ℵ‘𝑥) = 𝐴 ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
153, 14bitr2i 279 1 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2111  wrex 3107  wss 3881  ran crn 5520  Oncon0 6159   Fn wfn 6319  cfv 6324  ωcom 7560  cardccrd 9348  cale 9349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-oi 8958  df-har 9005  df-card 9352  df-aleph 9353
This theorem is referenced by:  iscard3  9504  alephinit  9506  cardinfima  9508  alephiso  9509  alephsson  9511  alephfp  9519
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