Theorem cardval2 9976

Description: An alternate version of the value of the cardinal number of a set. Compare cardval 10529. This theorem could be used to give a simpler definition of card in place of df-card 9924. 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 9929	. . . . . 6 ⊢ (card‘𝐴) ∈ On
2	1	oneli 6477	. . . . 5 ⊢ (𝑥 ∈ (card‘𝐴) → 𝑥 ∈ On)
3	2	pm4.71ri 569	. . . 4 ⊢ (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴)))
4		cardsdomel 9959	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥 ≺ 𝐴 ↔ 𝑥 ∈ (card‘𝐴)))
5	4	ancoms 463	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝑥 ≺ 𝐴 ↔ 𝑥 ∈ (card‘𝐴)))
6	5	pm5.32da 589	. . . 4 ⊢ (𝐴 ∈ dom card → ((𝑥 ∈ On ∧ 𝑥 ≺ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴))))
7	3, 6	bitr4id 293	. . 3 ⊢ (𝐴 ∈ dom card → (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴)))
8	7	eqabdv 2902	. 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴)})
9		df-rab 3424	. 2 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴)}
10	8, 9	eqtr4di 2822	1 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  {crab 3423   class class class wbr 5113  dom cdm 5662  Oncon0 6361  ‘cfv 6537   ≺ csdm 8941  cardccrd 9920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-card 9924

This theorem is referenced by:  ondomon  10546  alephsuc3  10564