Theorem cardval3 9881

Description: An alternate definition of the value of (card‘𝐴) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
cardval3	⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})

Distinct variable group:   𝑥,𝐴

Proof of Theorem cardval3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3465	. 2 ⊢ (𝐴 ∈ dom card → 𝐴 ∈ V)
2		isnum2 9874	. . . 4 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
3		rabn0 4348	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
4		intex 5294	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V)
5	2, 3, 4	3bitr2i 299	. . 3 ⊢ (𝐴 ∈ dom card ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V)
6	5	biimpi 216	. 2 ⊢ (𝐴 ∈ dom card → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V)
7		breq2 5106	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝑥 ≈ 𝐴))
8	7	rabbidv 3410	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦} = {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
9	8	inteqd 4911	. . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦} = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
10		df-card 9868	. . 3 ⊢ card = (𝑦 ∈ V ↦ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦})
11	9, 10	fvmptg 6948	. 2 ⊢ ((𝐴 ∈ V ∧ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V) → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
12	1, 6, 11	syl2anc 584	1 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  {crab 3402  Vcvv 3444  ∅c0 4292  ∩ cint 4906   class class class wbr 5102  dom cdm 5631  Oncon0 6320  ‘cfv 6499   ≈ cen 8892  cardccrd 9864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-en 8896  df-card 9868

This theorem is referenced by:  cardid2  9882  oncardval  9884  cardidm  9888  cardne  9894  cardval  10475