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Theorem isnum2 8981
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 8979 . . . 4 card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On
21fdmi 6213 . . 3 dom card = {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}
32eleq2i 2831 . 2 (𝐴 ∈ dom card ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦})
4 relen 8128 . . . . 5 Rel ≈
54brrelex2i 5316 . . . 4 (𝑥𝐴𝐴 ∈ V)
65rexlimivw 3167 . . 3 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
7 breq2 4808 . . . 4 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
87rexbidv 3190 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑥𝑦 ↔ ∃𝑥 ∈ On 𝑥𝐴))
96, 8elab3 3498 . 2 (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦} ↔ ∃𝑥 ∈ On 𝑥𝐴)
103, 9bitri 264 1 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  wcel 2139  {cab 2746  wrex 3051  Vcvv 3340   class class class wbr 4804  dom cdm 5266  Oncon0 5884  cen 8120  cardccrd 8971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-fun 6051  df-fn 6052  df-f 6053  df-en 8124  df-card 8975
This theorem is referenced by:  isnumi  8982  ennum  8983  xpnum  8987  cardval3  8988  dfac10c  9172  isfin7-2  9430  numth2  9505  inawinalem  9723
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