Theorem cardval2 10029

Description: An alternate version of the value of the cardinal number of a set. Compare cardval 10584. This theorem could be used to give a simpler definition of card in place of df-card 9977. 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 9982	. . . . . 6 ⊢ (card‘𝐴) ∈ On
2	1	oneli 6500	. . . . 5 ⊢ (𝑥 ∈ (card‘𝐴) → 𝑥 ∈ On)
3	2	pm4.71ri 560	. . . 4 ⊢ (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴)))
4		cardsdomel 10012	. . . . . 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 2873	. 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴)})
9		df-rab 3434	. 2 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴)}
10	8, 9	eqtr4di 2793	1 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  {crab 3433   class class class wbr 5148  dom cdm 5689  Oncon0 6386  ‘cfv 6563   ≺ csdm 8983  cardccrd 9973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-card 9977

This theorem is referenced by:  ondomon  10601  alephsuc3  10618