Theorem isnum2 9841

Description: A way to express well-orderability without bound or distinct variables. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
isnum2	⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem isnum2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardf2 9839	. . . 4 ⊢ card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On
2	1	fdmi 6663	. . 3 ⊢ dom card = {𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}
3	2	eleq2i 2820	. 2 ⊢ (𝐴 ∈ dom card ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦})
4		relen 8877	. . . . 5 ⊢ Rel ≈
5	4	brrelex2i 5676	. . . 4 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ∈ V)
6	5	rexlimivw 3126	. . 3 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → 𝐴 ∈ V)
7		breq2 5096	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝑥 ≈ 𝐴))
8	7	rexbidv 3153	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑥 ≈ 𝑦 ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴))
9	6, 8	elab3 3642	. 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦} ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
10	3, 9	bitri 275	1 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3436   class class class wbr 5092  dom cdm 5619  Oncon0 6307   ≈ cen 8869  cardccrd 9831

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 5235  ax-nul 5245  ax-pr 5371

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 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-fun 6484  df-fn 6485  df-f 6486  df-en 8873  df-card 9835

This theorem is referenced by:  isnumi  9842  ennum  9843  xpnum  9847  cardval3  9848  dfac10c  10033  isfin7-2  10290  numth2  10365  inawinalem  10583