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Theorem cardval2 10024
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 10579. This theorem could be used to give a simpler definition of card in place of df-card 9972. 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 9977 . . . . . 6 (cardβ€˜π΄) ∈ On
21oneli 6488 . . . . 5 (π‘₯ ∈ (cardβ€˜π΄) β†’ π‘₯ ∈ On)
32pm4.71ri 559 . . . 4 (π‘₯ ∈ (cardβ€˜π΄) ↔ (π‘₯ ∈ On ∧ π‘₯ ∈ (cardβ€˜π΄)))
4 cardsdomel 10007 . . . . . 6 ((π‘₯ ∈ On ∧ 𝐴 ∈ dom card) β†’ (π‘₯ β‰Ί 𝐴 ↔ π‘₯ ∈ (cardβ€˜π΄)))
54ancoms 457 . . . . 5 ((𝐴 ∈ dom card ∧ π‘₯ ∈ On) β†’ (π‘₯ β‰Ί 𝐴 ↔ π‘₯ ∈ (cardβ€˜π΄)))
65pm5.32da 577 . . . 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 2863 . 2 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) = {π‘₯ ∣ (π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴)})
9 df-rab 3431 . 2 {π‘₯ ∈ On ∣ π‘₯ β‰Ί 𝐴} = {π‘₯ ∣ (π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴)}
108, 9eqtr4di 2786 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 394   = wceq 1533   ∈ wcel 2098  {cab 2705  {crab 3430   class class class wbr 5152  dom cdm 5682  Oncon0 6374  β€˜cfv 6553   β‰Ί csdm 8971  cardccrd 9968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-er 8733  df-en 8973  df-dom 8974  df-sdom 8975  df-card 9972
This theorem is referenced by:  ondomon  10596  alephsuc3  10613
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