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Theorem isnum2 9104
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 9102 . . . 4 card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On
21fdmi 6301 . . 3 dom card = {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}
32eleq2i 2851 . 2 (𝐴 ∈ dom card ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦})
4 relen 8246 . . . . 5 Rel ≈
54brrelex2i 5407 . . . 4 (𝑥𝐴𝐴 ∈ V)
65rexlimivw 3211 . . 3 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
7 breq2 4890 . . . 4 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
87rexbidv 3237 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑥𝑦 ↔ ∃𝑥 ∈ On 𝑥𝐴))
96, 8elab3 3566 . 2 (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦} ↔ ∃𝑥 ∈ On 𝑥𝐴)
103, 9bitri 267 1 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1601  wcel 2107  {cab 2763  wrex 3091  Vcvv 3398   class class class wbr 4886  dom cdm 5355  Oncon0 5976  cen 8238  cardccrd 9094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-fun 6137  df-fn 6138  df-f 6139  df-en 8242  df-card 9098
This theorem is referenced by:  isnumi  9105  ennum  9106  xpnum  9110  cardval3  9111  dfac10c  9295  isfin7-2  9553  numth2  9628  inawinalem  9846
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