Theorem isnum2 9937

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 9935	. . . 4 ⊢ card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On
2	1	fdmi 6727	. . 3 ⊢ dom card = {𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}
3	2	eleq2i 2826	. 2 ⊢ (𝐴 ∈ dom card ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦})
4		relen 8941	. . . . 5 ⊢ Rel ≈
5	4	brrelex2i 5732	. . . 4 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ∈ V)
6	5	rexlimivw 3152	. . 3 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → 𝐴 ∈ V)
7		breq2 5152	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝑥 ≈ 𝐴))
8	7	rexbidv 3179	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑥 ≈ 𝑦 ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴))
9	6, 8	elab3 3676	. 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 205   = wceq 1542   ∈ wcel 2107  {cab 2710  ∃wrex 3071  Vcvv 3475   class class class wbr 5148  dom cdm 5676  Oncon0 6362   ≈ cen 8933  cardccrd 9927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-fun 6543  df-fn 6544  df-f 6545  df-en 8937  df-card 9931

This theorem is referenced by:  isnumi  9938  ennum  9939  xpnum  9943  cardval3  9944  dfac10c  10130  isfin7-2  10388  numth2  10463  inawinalem  10681