Theorem elno 27922

Description: Membership in the surreals. (Contributed by Scott Fenton, 11-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.) Avoid ax-rep 5232. (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 3471	. 2 ⊢ (𝐴 ∈ No → 𝐴 ∈ V)
2		vex 3454	. . . 4 ⊢ 𝑥 ∈ V
3		prex 5396	. . . 4 ⊢ {1o, 2o} ∈ V
4		fex2 7932	. . . 4 ⊢ ((𝐴:𝑥⟶{1o, 2o} ∧ 𝑥 ∈ V ∧ {1o, 2o} ∈ V) → 𝐴 ∈ V)
5	2, 3, 4	mp3an23 1482	. . 3 ⊢ (𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V)
6	5	rexlimivw 3159	. 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V)
7		feq1 6676	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝐴:𝑥⟶{1o, 2o}))
8	7	rexbidv 3186	. . 3 ⊢ (𝑓 = 𝐴 → (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
9		df-no 27919	. . 3 ⊢ No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}}
10	8, 9	elab2g 3634	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
11	1, 6, 10	pm5.21nii 381	1 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  Vcvv 3450  {cpr 4586  Oncon0 6352  ⟶wf 6524  1_oc1o 8448  2_oc2o 8449   No csur 27916

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5249  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-fun 6530  df-fn 6531  df-f 6532  df-no 27919

This theorem is used by:  nofun  27925  nodmon  27926  norn  27927  elno2  27930  noreson  27936