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Theorem iscard3 8799
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 8653 . . . . . . . . 9 (card‘𝐴) ∈ On
2 eleq1 2676 . . . . . . . . 9 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 222 . . . . . . . 8 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 eloni 5650 . . . . . . . 8 (𝐴 ∈ On → Ord 𝐴)
53, 4syl 17 . . . . . . 7 ((card‘𝐴) = 𝐴 → Ord 𝐴)
6 ordom 6966 . . . . . . 7 Ord ω
7 ordtri2or 5739 . . . . . . 7 ((Ord 𝐴 ∧ Ord ω) → (𝐴 ∈ ω ∨ ω ⊆ 𝐴))
85, 6, 7sylancl 693 . . . . . 6 ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ ω ⊆ 𝐴))
98ord 391 . . . . 5 ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → ω ⊆ 𝐴))
10 isinfcard 8798 . . . . . . 7 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
1110biimpi 205 . . . . . 6 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 ∈ ran ℵ)
1211expcom 450 . . . . 5 ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴𝐴 ∈ ran ℵ))
139, 12syld 46 . . . 4 ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → 𝐴 ∈ ran ℵ))
1413orrd 392 . . 3 ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
15 cardnn 8672 . . . 4 (𝐴 ∈ ω → (card‘𝐴) = 𝐴)
1610bicomi 213 . . . . 5 (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
1716simprbi 479 . . . 4 (𝐴 ∈ ran ℵ → (card‘𝐴) = 𝐴)
1815, 17jaoi 393 . . 3 ((𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ) → (card‘𝐴) = 𝐴)
1914, 18impbii 198 . 2 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
20 elun 3715 . 2 (𝐴 ∈ (ω ∪ ran ℵ) ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
2119, 20bitr4i 266 1 ((card‘𝐴) = 𝐴𝐴 ∈ (ω ∪ ran ℵ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wo 382  wa 383   = wceq 1475  wcel 1977  cun 3538  wss 3540  ran crn 5039  Ord word 5639  Oncon0 5640  cfv 5804  ωcom 6957  cardccrd 8644  cale 8645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-har 8346  df-card 8648  df-aleph 8649
This theorem is referenced by:  cardnum  8800  carduniima  8802  cardinfima  8803  cfpwsdom  9285  gch2  9376
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