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.

Step	Hyp	Ref	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)
5	2, 3, 4	mp3an23 1455	. . 3 ⊢ (𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V)
6	5	rexlimivw 3151	. 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V)
7		feq1 6716	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝐴:𝑥⟶{1o, 2o}))
8	7	rexbidv 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}}
10	8, 9	elab2g 3680	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
11	1, 6, 10	pm5.21nii 378	1 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  Vcvv 3480  {cpr 4628  Oncon0 6384  ⟶wf 6557  1_oc1o 8499  2_oc2o 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