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Theorem elno 32388
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 3413 . 2 (𝐴 No 𝐴 ∈ V)
2 fex 6761 . . . 4 ((𝐴:𝑥⟶{1o, 2o} ∧ 𝑥 ∈ On) → 𝐴 ∈ V)
32ancoms 452 . . 3 ((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) → 𝐴 ∈ V)
43rexlimiva 3209 . 2 (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V)
5 feq1 6272 . . . 4 (𝑓 = 𝐴 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝐴:𝑥⟶{1o, 2o}))
65rexbidv 3236 . . 3 (𝑓 = 𝐴 → (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
7 df-no 32385 . . 3 No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}}
86, 7elab2g 3560 . 2 (𝐴 ∈ V → (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
91, 4, 8pm5.21nii 370 1 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1601  wcel 2106  wrex 3090  Vcvv 3397  {cpr 4399  Oncon0 5976  wf 6131  1oc1o 7836  2oc2o 7837   No csur 32382
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 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  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-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-no 32385
This theorem is referenced by:  nofun  32391  nodmon  32392  norn  32393  elno2  32396  noreson  32402
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