Theorem cardval2 10032

Description: An alternate version of the value of the cardinal number of a set. Compare cardval 10587. This theorem could be used to give a simpler definition of card in place of df-card 9980. 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 ∣ 𝑥 ≺ 𝐴})

Distinct variable group:   𝑥,𝐴

Proof of Theorem cardval2

Step	Hyp	Ref	Expression
1		cardon 9985	. . . . . 6 ⊢ (card‘𝐴) ∈ On
2	1	oneli 6497	. . . . 5 ⊢ (𝑥 ∈ (card‘𝐴) → 𝑥 ∈ On)
3	2	pm4.71ri 560	. . . 4 ⊢ (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴)))
4		cardsdomel 10015	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥 ≺ 𝐴 ↔ 𝑥 ∈ (card‘𝐴)))
5	4	ancoms 458	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝑥 ≺ 𝐴 ↔ 𝑥 ∈ (card‘𝐴)))
6	5	pm5.32da 579	. . . 4 ⊢ (𝐴 ∈ dom card → ((𝑥 ∈ On ∧ 𝑥 ≺ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴))))
7	3, 6	bitr4id 290	. . 3 ⊢ (𝐴 ∈ dom card → (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴)))
8	7	eqabdv 2874	. 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴)})
9		df-rab 3436	. 2 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴)}
10	8, 9	eqtr4di 2794	1 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2713  {crab 3435   class class class wbr 5142  dom cdm 5684  Oncon0 6383  ‘cfv 6560   ≺ csdm 8985  cardccrd 9976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-card 9980

This theorem is referenced by:  ondomon  10604  alephsuc3  10621