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Theorem isnum2 9360
 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 9358 . . . 4 card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On
21fdmi 6498 . . 3 dom card = {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}
32eleq2i 2881 . 2 (𝐴 ∈ dom card ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦})
4 relen 8499 . . . . 5 Rel ≈
54brrelex2i 5573 . . . 4 (𝑥𝐴𝐴 ∈ V)
65rexlimivw 3241 . . 3 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
7 breq2 5034 . . . 4 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
87rexbidv 3256 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑥𝑦 ↔ ∃𝑥 ∈ On 𝑥𝐴))
96, 8elab3 3622 . 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 1538   ∈ wcel 2111  {cab 2776  ∃wrex 3107  Vcvv 3441   class class class wbr 5030  dom cdm 5519  Oncon0 6159   ≈ cen 8491  cardccrd 9350 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-fun 6326  df-fn 6327  df-f 6328  df-en 8495  df-card 9354 This theorem is referenced by:  isnumi  9361  ennum  9362  xpnum  9366  cardval3  9367  dfac10c  9551  isfin7-2  9809  numth2  9884  inawinalem  10102
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