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Theorem cardval2 8677
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 9224. This theorem could be used to give us a simpler definition of card in place of df-card 8625. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)
Assertion
Ref Expression
cardval2 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
Distinct variable group:   𝑥,𝐴

Proof of Theorem cardval2
StepHypRef Expression
1 cardsdomel 8660 . . . . . 6 ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
21ancoms 467 . . . . 5 ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
32pm5.32da 670 . . . 4 (𝐴 ∈ dom card → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴))))
4 cardon 8630 . . . . . 6 (card‘𝐴) ∈ On
54oneli 5738 . . . . 5 (𝑥 ∈ (card‘𝐴) → 𝑥 ∈ On)
65pm4.71ri 662 . . . 4 (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴)))
73, 6syl6rbbr 277 . . 3 (𝐴 ∈ dom card → (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥𝐴)))
87abbi2dv 2728 . 2 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)})
9 df-rab 2904 . 2 {𝑥 ∈ On ∣ 𝑥𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
108, 9syl6eqr 2661 1 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {cab 2595  {crab 2899   class class class wbr 4577  dom cdm 5028  Oncon0 5626  cfv 5790  csdm 7817  cardccrd 8621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-card 8625
This theorem is referenced by:  ondomon  9241  alephsuc3  9258
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