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Theorem iscard 9398
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 9367 . . 3 (card‘𝐴) ∈ On
2 eleq1 2905 . . 3 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 234 . 2 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 cardonle 9380 . . . 4 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
5 eqss 3986 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
65baibr 537 . . . 4 ((card‘𝐴) ⊆ 𝐴 → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
74, 6syl 17 . . 3 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
8 onelon 6215 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
9 onenon 9372 . . . . . . 7 (𝐴 ∈ On → 𝐴 ∈ dom card)
109adantr 481 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
11 cardsdomel 9397 . . . . . 6 ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
128, 10, 11syl2anc 584 . . . . 5 ((𝐴 ∈ On ∧ 𝑥𝐴) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
1312ralbidva 3201 . . . 4 (𝐴 ∈ On → (∀𝑥𝐴 𝑥𝐴 ↔ ∀𝑥𝐴 𝑥 ∈ (card‘𝐴)))
14 dfss3 3960 . . . 4 (𝐴 ⊆ (card‘𝐴) ↔ ∀𝑥𝐴 𝑥 ∈ (card‘𝐴))
1513, 14syl6rbbr 291 . . 3 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ∀𝑥𝐴 𝑥𝐴))
167, 15bitr3d 282 . 2 (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴))
173, 16biadanii 819 1 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥𝐴 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  wss 3940   class class class wbr 5063  dom cdm 5554  Oncon0 6190  cfv 6354  csdm 8502  cardccrd 9358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-card 9362
This theorem is referenced by:  cardprclem  9402  cardmin2  9421  infxpenlem  9433  alephsuc2  9500  cardmin  9980  alephreg  9998  pwcfsdom  9999  winalim2  10112  gchina  10115  inar1  10191  r1tskina  10198  gruina  10234  iscard5  39785
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