Theorem isnum2 9983

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 9981	. . . 4 ⊢ card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On
2	1	fdmi 6748	. . 3 ⊢ dom card = {𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}
3	2	eleq2i 2831	. 2 ⊢ (𝐴 ∈ dom card ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦})
4		relen 8989	. . . . 5 ⊢ Rel ≈
5	4	brrelex2i 5746	. . . 4 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ∈ V)
6	5	rexlimivw 3149	. . 3 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → 𝐴 ∈ V)
7		breq2 5152	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝑥 ≈ 𝐴))
8	7	rexbidv 3177	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑥 ≈ 𝑦 ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴))
9	6, 8	elab3 3689	. 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 1537   ∈ wcel 2106  {cab 2712  ∃wrex 3068  Vcvv 3478   class class class wbr 5148  dom cdm 5689  Oncon0 6386   ≈ cen 8981  cardccrd 9973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-fun 6565  df-fn 6566  df-f 6567  df-en 8985  df-card 9977

This theorem is referenced by:  isnumi  9984  ennum  9985  xpnum  9989  cardval3  9990  dfac10c  10177  isfin7-2  10434  numth2  10509  inawinalem  10727