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Theorem iscard 8752
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 8721 . . 3 (card‘𝐴) ∈ On
2 eleq1 2686 . . 3 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 223 . 2 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 cardonle 8734 . . . 4 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
5 eqss 3602 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
65baibr 944 . . . 4 ((card‘𝐴) ⊆ 𝐴 → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
74, 6syl 17 . . 3 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
8 onelon 5712 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
9 onenon 8726 . . . . . . 7 (𝐴 ∈ On → 𝐴 ∈ dom card)
109adantr 481 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
11 cardsdomel 8751 . . . . . 6 ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
128, 10, 11syl2anc 692 . . . . 5 ((𝐴 ∈ On ∧ 𝑥𝐴) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
1312ralbidva 2980 . . . 4 (𝐴 ∈ On → (∀𝑥𝐴 𝑥𝐴 ↔ ∀𝑥𝐴 𝑥 ∈ (card‘𝐴)))
14 dfss3 3577 . . . 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 673 1 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥𝐴 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3559   class class class wbr 4618  dom cdm 5079  Oncon0 5687  cfv 5852  csdm 7905  cardccrd 8712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-card 8716
This theorem is referenced by:  cardprclem  8756  cardmin2  8775  infxpenlem  8787  alephsuc2  8854  cardmin  9337  alephreg  9355  pwcfsdom  9356  winalim2  9469  gchina  9472  inar1  9548  r1tskina  9555  gruina  9591
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