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Theorem cardval2 9988
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 10543. This theorem could be used to give a simpler definition of card in place of df-card 9936. 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 9941 . . . . . 6 (cardβ€˜π΄) ∈ On
21oneli 6472 . . . . 5 (π‘₯ ∈ (cardβ€˜π΄) β†’ π‘₯ ∈ On)
32pm4.71ri 560 . . . 4 (π‘₯ ∈ (cardβ€˜π΄) ↔ (π‘₯ ∈ On ∧ π‘₯ ∈ (cardβ€˜π΄)))
4 cardsdomel 9971 . . . . . 6 ((π‘₯ ∈ On ∧ 𝐴 ∈ dom card) β†’ (π‘₯ β‰Ί 𝐴 ↔ π‘₯ ∈ (cardβ€˜π΄)))
54ancoms 458 . . . . 5 ((𝐴 ∈ dom card ∧ π‘₯ ∈ On) β†’ (π‘₯ β‰Ί 𝐴 ↔ π‘₯ ∈ (cardβ€˜π΄)))
65pm5.32da 578 . . . 4 (𝐴 ∈ dom card β†’ ((π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴) ↔ (π‘₯ ∈ On ∧ π‘₯ ∈ (cardβ€˜π΄))))
73, 6bitr4id 290 . . 3 (𝐴 ∈ dom card β†’ (π‘₯ ∈ (cardβ€˜π΄) ↔ (π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴)))
87eqabdv 2861 . 2 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) = {π‘₯ ∣ (π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴)})
9 df-rab 3427 . 2 {π‘₯ ∈ On ∣ π‘₯ β‰Ί 𝐴} = {π‘₯ ∣ (π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴)}
108, 9eqtr4di 2784 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 395   = wceq 1533   ∈ wcel 2098  {cab 2703  {crab 3426   class class class wbr 5141  dom cdm 5669  Oncon0 6358  β€˜cfv 6537   β‰Ί csdm 8940  cardccrd 9932
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-card 9936
This theorem is referenced by:  ondomon  10560  alephsuc3  10577
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