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Theorem iscard 9427
 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 9396 . . 3 (card‘𝐴) ∈ On
2 eleq1 2840 . . 3 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 236 . 2 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 cardonle 9409 . . . 4 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
5 eqss 3908 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
65baibr 541 . . . 4 ((card‘𝐴) ⊆ 𝐴 → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
74, 6syl 17 . . 3 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
8 dfss3 3881 . . . 4 (𝐴 ⊆ (card‘𝐴) ↔ ∀𝑥𝐴 𝑥 ∈ (card‘𝐴))
9 onelon 6192 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
10 onenon 9401 . . . . . . 7 (𝐴 ∈ On → 𝐴 ∈ dom card)
1110adantr 485 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
12 cardsdomel 9426 . . . . . 6 ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
139, 11, 12syl2anc 588 . . . . 5 ((𝐴 ∈ On ∧ 𝑥𝐴) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
1413ralbidva 3126 . . . 4 (𝐴 ∈ On → (∀𝑥𝐴 𝑥𝐴 ↔ ∀𝑥𝐴 𝑥 ∈ (card‘𝐴)))
158, 14bitr4id 294 . . 3 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ∀𝑥𝐴 𝑥𝐴))
167, 15bitr3d 284 . 2 (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴))
173, 16biadanii 822 1 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥𝐴 𝑥𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071   ⊆ wss 3859   class class class wbr 5030  dom cdm 5522  Oncon0 6167  ‘cfv 6333   ≺ csdm 8524  cardccrd 9387 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-tp 4525  df-op 4527  df-uni 4797  df-int 4837  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-ord 6170  df-on 6171  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-er 8297  df-en 8526  df-dom 8527  df-sdom 8528  df-card 9391 This theorem is referenced by:  cardprclem  9431  cardmin2  9451  infxpenlem  9463  alephsuc2  9530  cardmin  10014  alephreg  10032  pwcfsdom  10033  winalim2  10146  gchina  10149  inar1  10225  r1tskina  10232  gruina  10268  iscard5  40605
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