Theorem isnum2 9860

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 9858	. . . 4 ⊢ card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On
2	1	fdmi 6673	. . 3 ⊢ dom card = {𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}
3	2	eleq2i 2829	. 2 ⊢ (𝐴 ∈ dom card ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦})
4		relen 8891	. . . . 5 ⊢ Rel ≈
5	4	brrelex2i 5681	. . . 4 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ∈ V)
6	5	rexlimivw 3135	. . 3 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → 𝐴 ∈ V)
7		breq2 5090	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝑥 ≈ 𝐴))
8	7	rexbidv 3162	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑥 ≈ 𝑦 ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴))
9	6, 8	elab3 3630	. 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 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3430   class class class wbr 5086  dom cdm 5624  Oncon0 6317   ≈ cen 8883  cardccrd 9850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-fun 6494  df-fn 6495  df-f 6496  df-en 8887  df-card 9854

This theorem is referenced by:  isnumi  9861  ennum  9862  xpnum  9866  cardval3  9867  dfac10c  10052  isfin7-2  10309  numth2  10384  inawinalem  10603