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Theorem cardval2 9985
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 10540. This theorem could be used to give a simpler definition of card in place of df-card 9933. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)
Assertion
Ref Expression
cardval2 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) = {π‘₯ ∈ On ∣ π‘₯ β‰Ί 𝐴})
Distinct variable group:   π‘₯,𝐴

Proof of Theorem cardval2
StepHypRef Expression
1 cardon 9938 . . . . . 6 (cardβ€˜π΄) ∈ On
21oneli 6478 . . . . 5 (π‘₯ ∈ (cardβ€˜π΄) β†’ π‘₯ ∈ On)
32pm4.71ri 561 . . . 4 (π‘₯ ∈ (cardβ€˜π΄) ↔ (π‘₯ ∈ On ∧ π‘₯ ∈ (cardβ€˜π΄)))
4 cardsdomel 9968 . . . . . 6 ((π‘₯ ∈ On ∧ 𝐴 ∈ dom card) β†’ (π‘₯ β‰Ί 𝐴 ↔ π‘₯ ∈ (cardβ€˜π΄)))
54ancoms 459 . . . . 5 ((𝐴 ∈ dom card ∧ π‘₯ ∈ On) β†’ (π‘₯ β‰Ί 𝐴 ↔ π‘₯ ∈ (cardβ€˜π΄)))
65pm5.32da 579 . . . 4 (𝐴 ∈ dom card β†’ ((π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴) ↔ (π‘₯ ∈ On ∧ π‘₯ ∈ (cardβ€˜π΄))))
73, 6bitr4id 289 . . 3 (𝐴 ∈ dom card β†’ (π‘₯ ∈ (cardβ€˜π΄) ↔ (π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴)))
87eqabdv 2867 . 2 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) = {π‘₯ ∣ (π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴)})
9 df-rab 3433 . 2 {π‘₯ ∈ On ∣ π‘₯ β‰Ί 𝐴} = {π‘₯ ∣ (π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴)}
108, 9eqtr4di 2790 1 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) = {π‘₯ ∈ On ∣ π‘₯ β‰Ί 𝐴})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  {crab 3432   class class class wbr 5148  dom cdm 5676  Oncon0 6364  β€˜cfv 6543   β‰Ί csdm 8937  cardccrd 9929
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-card 9933
This theorem is referenced by:  ondomon  10557  alephsuc3  10574
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