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Theorem iscard 9966
Description: Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
iscard ((cardβ€˜π΄) = 𝐴 ↔ (𝐴 ∈ On ∧ βˆ€π‘₯ ∈ 𝐴 π‘₯ β‰Ί 𝐴))
Distinct variable group:   π‘₯,𝐴

Proof of Theorem iscard
StepHypRef Expression
1 cardon 9935 . . 3 (cardβ€˜π΄) ∈ On
2 eleq1 2821 . . 3 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 232 . 2 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
4 cardonle 9948 . . . 4 (𝐴 ∈ On β†’ (cardβ€˜π΄) βŠ† 𝐴)
5 eqss 3996 . . . . 5 ((cardβ€˜π΄) = 𝐴 ↔ ((cardβ€˜π΄) βŠ† 𝐴 ∧ 𝐴 βŠ† (cardβ€˜π΄)))
65baibr 537 . . . 4 ((cardβ€˜π΄) βŠ† 𝐴 β†’ (𝐴 βŠ† (cardβ€˜π΄) ↔ (cardβ€˜π΄) = 𝐴))
74, 6syl 17 . . 3 (𝐴 ∈ On β†’ (𝐴 βŠ† (cardβ€˜π΄) ↔ (cardβ€˜π΄) = 𝐴))
8 dfss3 3969 . . . 4 (𝐴 βŠ† (cardβ€˜π΄) ↔ βˆ€π‘₯ ∈ 𝐴 π‘₯ ∈ (cardβ€˜π΄))
9 onelon 6386 . . . . . 6 ((𝐴 ∈ On ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ On)
10 onenon 9940 . . . . . . 7 (𝐴 ∈ On β†’ 𝐴 ∈ dom card)
1110adantr 481 . . . . . 6 ((𝐴 ∈ On ∧ π‘₯ ∈ 𝐴) β†’ 𝐴 ∈ dom card)
12 cardsdomel 9965 . . . . . 6 ((π‘₯ ∈ On ∧ 𝐴 ∈ dom card) β†’ (π‘₯ β‰Ί 𝐴 ↔ π‘₯ ∈ (cardβ€˜π΄)))
139, 11, 12syl2anc 584 . . . . 5 ((𝐴 ∈ On ∧ π‘₯ ∈ 𝐴) β†’ (π‘₯ β‰Ί 𝐴 ↔ π‘₯ ∈ (cardβ€˜π΄)))
1413ralbidva 3175 . . . 4 (𝐴 ∈ On β†’ (βˆ€π‘₯ ∈ 𝐴 π‘₯ β‰Ί 𝐴 ↔ βˆ€π‘₯ ∈ 𝐴 π‘₯ ∈ (cardβ€˜π΄)))
158, 14bitr4id 289 . . 3 (𝐴 ∈ On β†’ (𝐴 βŠ† (cardβ€˜π΄) ↔ βˆ€π‘₯ ∈ 𝐴 π‘₯ β‰Ί 𝐴))
167, 15bitr3d 280 . 2 (𝐴 ∈ On β†’ ((cardβ€˜π΄) = 𝐴 ↔ βˆ€π‘₯ ∈ 𝐴 π‘₯ β‰Ί 𝐴))
173, 16biadanii 820 1 ((cardβ€˜π΄) = 𝐴 ↔ (𝐴 ∈ On ∧ βˆ€π‘₯ ∈ 𝐴 π‘₯ β‰Ί 𝐴))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3947   class class class wbr 5147  dom cdm 5675  Oncon0 6361  β€˜cfv 6540   β‰Ί csdm 8934  cardccrd 9926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-card 9930
This theorem is referenced by:  cardprclem  9970  cardmin2  9990  infxpenlem  10004  alephsuc2  10071  cardmin  10555  alephreg  10573  pwcfsdom  10574  winalim2  10687  gchina  10690  inar1  10766  r1tskina  10773  gruina  10809  iscard5  42272
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