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Theorem iscard3 9596
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 9449 . . . . . . . . 9 (card‘𝐴) ∈ On
2 eleq1 2821 . . . . . . . . 9 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 236 . . . . . . . 8 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 eloni 6183 . . . . . . . 8 (𝐴 ∈ On → Ord 𝐴)
53, 4syl 17 . . . . . . 7 ((card‘𝐴) = 𝐴 → Ord 𝐴)
6 ordom 7611 . . . . . . 7 Ord ω
7 ordtri2or 6268 . . . . . . 7 ((Ord 𝐴 ∧ Ord ω) → (𝐴 ∈ ω ∨ ω ⊆ 𝐴))
85, 6, 7sylancl 589 . . . . . 6 ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ ω ⊆ 𝐴))
98ord 863 . . . . 5 ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → ω ⊆ 𝐴))
10 isinfcard 9595 . . . . . . 7 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
1110biimpi 219 . . . . . 6 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 ∈ ran ℵ)
1211expcom 417 . . . . 5 ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴𝐴 ∈ ran ℵ))
139, 12syld 47 . . . 4 ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → 𝐴 ∈ ran ℵ))
1413orrd 862 . . 3 ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
15 cardnn 9468 . . . 4 (𝐴 ∈ ω → (card‘𝐴) = 𝐴)
1610bicomi 227 . . . . 5 (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
1716simprbi 500 . . . 4 (𝐴 ∈ ran ℵ → (card‘𝐴) = 𝐴)
1815, 17jaoi 856 . . 3 ((𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ) → (card‘𝐴) = 𝐴)
1914, 18impbii 212 . 2 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
20 elun 4040 . 2 (𝐴 ∈ (ω ∪ ran ℵ) ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
2119, 20bitr4i 281 1 ((card‘𝐴) = 𝐴𝐴 ∈ (ω ∪ ran ℵ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399  wo 846   = wceq 1542  wcel 2114  cun 3842  wss 3844  ran crn 5527  Ord word 6172  Oncon0 6173  cfv 6340  ωcom 7602  cardccrd 9440  cale 9441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482  ax-inf2 9180
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-int 4838  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-se 5485  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-isom 6349  df-riota 7130  df-om 7603  df-wrecs 7979  df-recs 8040  df-rdg 8078  df-er 8323  df-en 8559  df-dom 8560  df-sdom 8561  df-fin 8562  df-oi 9050  df-har 9097  df-card 9444  df-aleph 9445
This theorem is referenced by:  cardnum  9597  carduniima  9599  cardinfima  9600  cfpwsdom  10087  gch2  10178
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