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Theorem iscard3 9511
 Description: Two ways to express the property of being a cardinal number. (Contributed by NM, 9-Nov-2003.)
Assertion
Ref Expression
iscard3 ((card‘𝐴) = 𝐴𝐴 ∈ (ω ∪ ran ℵ))

Proof of Theorem iscard3
StepHypRef Expression
1 cardon 9365 . . . . . . . . 9 (card‘𝐴) ∈ On
2 eleq1 2898 . . . . . . . . 9 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 235 . . . . . . . 8 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 eloni 6194 . . . . . . . 8 (𝐴 ∈ On → Ord 𝐴)
53, 4syl 17 . . . . . . 7 ((card‘𝐴) = 𝐴 → Ord 𝐴)
6 ordom 7581 . . . . . . 7 Ord ω
7 ordtri2or 6279 . . . . . . 7 ((Ord 𝐴 ∧ Ord ω) → (𝐴 ∈ ω ∨ ω ⊆ 𝐴))
85, 6, 7sylancl 588 . . . . . 6 ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ ω ⊆ 𝐴))
98ord 860 . . . . 5 ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → ω ⊆ 𝐴))
10 isinfcard 9510 . . . . . . 7 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
1110biimpi 218 . . . . . 6 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 ∈ ran ℵ)
1211expcom 416 . . . . 5 ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴𝐴 ∈ ran ℵ))
139, 12syld 47 . . . 4 ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → 𝐴 ∈ ran ℵ))
1413orrd 859 . . 3 ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
15 cardnn 9384 . . . 4 (𝐴 ∈ ω → (card‘𝐴) = 𝐴)
1610bicomi 226 . . . . 5 (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
1716simprbi 499 . . . 4 (𝐴 ∈ ran ℵ → (card‘𝐴) = 𝐴)
1815, 17jaoi 853 . . 3 ((𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ) → (card‘𝐴) = 𝐴)
1914, 18impbii 211 . 2 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
20 elun 4123 . 2 (𝐴 ∈ (ω ∪ ran ℵ) ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
2119, 20bitr4i 280 1 ((card‘𝐴) = 𝐴𝐴 ∈ (ω ∪ ran ℵ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1531   ∈ wcel 2108   ∪ cun 3932   ⊆ wss 3934  ran crn 5549  Ord word 6183  Oncon0 6184  ‘cfv 6348  ωcom 7572  cardccrd 9356  ℵcale 9357 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-oi 8966  df-har 9014  df-card 9360  df-aleph 9361 This theorem is referenced by:  cardnum  9512  carduniima  9514  cardinfima  9515  cfpwsdom  9998  gch2  10089
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