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Theorem iscard 9001
Description: Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
iscard ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥𝐴 𝑥𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem iscard
StepHypRef Expression
1 cardon 8970 . . 3 (card‘𝐴) ∈ On
2 eleq1 2838 . . 3 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 223 . 2 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 cardonle 8983 . . . 4 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
5 eqss 3767 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
65baibr 526 . . . 4 ((card‘𝐴) ⊆ 𝐴 → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
74, 6syl 17 . . 3 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
8 onelon 5891 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
9 onenon 8975 . . . . . . 7 (𝐴 ∈ On → 𝐴 ∈ dom card)
109adantr 466 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
11 cardsdomel 9000 . . . . . 6 ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
128, 10, 11syl2anc 573 . . . . 5 ((𝐴 ∈ On ∧ 𝑥𝐴) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
1312ralbidva 3134 . . . 4 (𝐴 ∈ On → (∀𝑥𝐴 𝑥𝐴 ↔ ∀𝑥𝐴 𝑥 ∈ (card‘𝐴)))
14 dfss3 3741 . . . 4 (𝐴 ⊆ (card‘𝐴) ↔ ∀𝑥𝐴 𝑥 ∈ (card‘𝐴))
1513, 14syl6rbbr 279 . . 3 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ∀𝑥𝐴 𝑥𝐴))
167, 15bitr3d 270 . 2 (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴))
173, 16biadan2 820 1 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥𝐴 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wss 3723   class class class wbr 4786  dom cdm 5249  Oncon0 5866  cfv 6031  csdm 8108  cardccrd 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-card 8965
This theorem is referenced by:  cardprclem  9005  cardmin2  9024  infxpenlem  9036  alephsuc2  9103  cardmin  9588  alephreg  9606  pwcfsdom  9607  winalim2  9720  gchina  9723  inar1  9799  r1tskina  9806  gruina  9842
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