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Theorem isnum2 9561
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 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem isnum2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardf2 9559 . . . 4 card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On
21fdmi 6557 . . 3 dom card = {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}
32eleq2i 2829 . 2 (𝐴 ∈ dom card ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦})
4 relen 8631 . . . . 5 Rel ≈
54brrelex2i 5606 . . . 4 (𝑥𝐴𝐴 ∈ V)
65rexlimivw 3201 . . 3 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
7 breq2 5057 . . . 4 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
87rexbidv 3216 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑥𝑦 ↔ ∃𝑥 ∈ On 𝑥𝐴))
96, 8elab3 3595 . 2 (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦} ↔ ∃𝑥 ∈ On 𝑥𝐴)
103, 9bitri 278 1 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1543  wcel 2110  {cab 2714  wrex 3062  Vcvv 3408   class class class wbr 5053  dom cdm 5551  Oncon0 6213  cen 8623  cardccrd 9551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-fun 6382  df-fn 6383  df-f 6384  df-en 8627  df-card 9555
This theorem is referenced by:  isnumi  9562  ennum  9563  xpnum  9567  cardval3  9568  dfac10c  9752  isfin7-2  10010  numth2  10085  inawinalem  10303
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