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Theorem elno 27566
Description: Membership in the surreals. (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
Assertion
Ref Expression
elno (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
Distinct variable group:   𝑥,𝐴

Proof of Theorem elno
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 elex 3488 . 2 (𝐴 No 𝐴 ∈ V)
2 fex 7232 . . . 4 ((𝐴:𝑥⟶{1o, 2o} ∧ 𝑥 ∈ On) → 𝐴 ∈ V)
32ancoms 458 . . 3 ((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) → 𝐴 ∈ V)
43rexlimiva 3142 . 2 (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V)
5 feq1 6697 . . . 4 (𝑓 = 𝐴 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝐴:𝑥⟶{1o, 2o}))
65rexbidv 3173 . . 3 (𝑓 = 𝐴 → (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
7 df-no 27563 . . 3 No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}}
86, 7elab2g 3667 . 2 (𝐴 ∈ V → (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
91, 4, 8pm5.21nii 378 1 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wcel 2099  wrex 3065  Vcvv 3469  {cpr 4626  Oncon0 6363  wf 6538  1oc1o 8473  2oc2o 8474   No csur 27560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-no 27563
This theorem is referenced by:  nofun  27569  nodmon  27570  norn  27571  elno2  27574  noreson  27580
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