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Theorem iscard 8684
 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 8653 . . 3 (card‘𝐴) ∈ On
2 eleq1 2676 . . 3 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 222 . 2 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 cardonle 8666 . . . 4 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
5 eqss 3583 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
65baibr 943 . . . 4 ((card‘𝐴) ⊆ 𝐴 → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
74, 6syl 17 . . 3 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
8 onelon 5665 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
9 onenon 8658 . . . . . . 7 (𝐴 ∈ On → 𝐴 ∈ dom card)
109adantr 480 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
11 cardsdomel 8683 . . . . . 6 ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
128, 10, 11syl2anc 691 . . . . 5 ((𝐴 ∈ On ∧ 𝑥𝐴) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
1312ralbidva 2968 . . . 4 (𝐴 ∈ On → (∀𝑥𝐴 𝑥𝐴 ↔ ∀𝑥𝐴 𝑥 ∈ (card‘𝐴)))
14 dfss3 3558 . . . 4 (𝐴 ⊆ (card‘𝐴) ↔ ∀𝑥𝐴 𝑥 ∈ (card‘𝐴))
1513, 14syl6rbbr 278 . . 3 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ∀𝑥𝐴 𝑥𝐴))
167, 15bitr3d 269 . 2 (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴))
173, 16biadan2 672 1 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥𝐴 𝑥𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038  Oncon0 5640  ‘cfv 5804   ≺ csdm 7840  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-card 8648 This theorem is referenced by:  cardprclem  8688  cardmin2  8707  infxpenlem  8719  alephsuc2  8786  cardmin  9265  alephreg  9283  pwcfsdom  9284  winalim2  9397  gchina  9400  inar1  9476  r1tskina  9483  gruina  9519
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