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Theorem iscard3 9236
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 9090 . . . . . . . . 9 (card‘𝐴) ∈ On
2 eleq1 2894 . . . . . . . . 9 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 225 . . . . . . . 8 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 eloni 5977 . . . . . . . 8 (𝐴 ∈ On → Ord 𝐴)
53, 4syl 17 . . . . . . 7 ((card‘𝐴) = 𝐴 → Ord 𝐴)
6 ordom 7340 . . . . . . 7 Ord ω
7 ordtri2or 6062 . . . . . . 7 ((Ord 𝐴 ∧ Ord ω) → (𝐴 ∈ ω ∨ ω ⊆ 𝐴))
85, 6, 7sylancl 580 . . . . . 6 ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ ω ⊆ 𝐴))
98ord 895 . . . . 5 ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → ω ⊆ 𝐴))
10 isinfcard 9235 . . . . . . 7 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
1110biimpi 208 . . . . . 6 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 ∈ ran ℵ)
1211expcom 404 . . . . 5 ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴𝐴 ∈ ran ℵ))
139, 12syld 47 . . . 4 ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → 𝐴 ∈ ran ℵ))
1413orrd 894 . . 3 ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
15 cardnn 9109 . . . 4 (𝐴 ∈ ω → (card‘𝐴) = 𝐴)
1610bicomi 216 . . . . 5 (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
1716simprbi 492 . . . 4 (𝐴 ∈ ran ℵ → (card‘𝐴) = 𝐴)
1815, 17jaoi 888 . . 3 ((𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ) → (card‘𝐴) = 𝐴)
1914, 18impbii 201 . 2 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
20 elun 3982 . 2 (𝐴 ∈ (ω ∪ ran ℵ) ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
2119, 20bitr4i 270 1 ((card‘𝐴) = 𝐴𝐴 ∈ (ω ∪ ran ℵ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wa 386  wo 878   = wceq 1656  wcel 2164  cun 3796  wss 3798  ran crn 5347  Ord word 5966  Oncon0 5967  cfv 6127  ωcom 7331  cardccrd 9081  cale 9082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-om 7332  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-oi 8691  df-har 8739  df-card 9085  df-aleph 9086
This theorem is referenced by:  cardnum  9237  carduniima  9239  cardinfima  9240  cfpwsdom  9728  gch2  9819
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