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Theorem elno 27690
Description: Membership in the surreals. (Contributed by Scott Fenton, 11-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.) Avoid ax-rep 5279. (Revised by SN, 5-Jun-2025.)
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 3501 . 2 (𝐴 No 𝐴 ∈ V)
2 vex 3484 . . . 4 𝑥 ∈ V
3 prex 5437 . . . 4 {1o, 2o} ∈ V
4 fex2 7958 . . . 4 ((𝐴:𝑥⟶{1o, 2o} ∧ 𝑥 ∈ V ∧ {1o, 2o} ∈ V) → 𝐴 ∈ V)
52, 3, 4mp3an23 1455 . . 3 (𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V)
65rexlimivw 3151 . 2 (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V)
7 feq1 6716 . . . 4 (𝑓 = 𝐴 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝐴:𝑥⟶{1o, 2o}))
87rexbidv 3179 . . 3 (𝑓 = 𝐴 → (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
9 df-no 27687 . . 3 No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}}
108, 9elab2g 3680 . 2 (𝐴 ∈ V → (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
111, 6, 10pm5.21nii 378 1 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  {cpr 4628  Oncon0 6384  wf 6557  1oc1o 8499  2oc2o 8500   No csur 27684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-no 27687
This theorem is referenced by:  nofun  27694  nodmon  27695  norn  27696  elno2  27699  noreson  27705
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