Theorem elno 33266

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.

Step	Hyp	Ref	Expression
1		elex 3459	. 2 ⊢ (𝐴 ∈ No → 𝐴 ∈ V)
2		fex 6966	. . . 4 ⊢ ((𝐴:𝑥⟶{1o, 2o} ∧ 𝑥 ∈ On) → 𝐴 ∈ V)
3	2	ancoms 462	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) → 𝐴 ∈ V)
4	3	rexlimiva 3240	. 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V)
5		feq1 6468	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝐴:𝑥⟶{1o, 2o}))
6	5	rexbidv 3256	. . 3 ⊢ (𝑓 = 𝐴 → (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
7		df-no 33263	. . 3 ⊢ No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}}
8	6, 7	elab2g 3616	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
9	1, 4, 8	pm5.21nii 383	1 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})

Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441  {cpr 4527  Oncon0 6159  ⟶wf 6320  1_oc1o 8078  2_oc2o 8079   No csur 33260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-no 33263

This theorem is referenced by:  nofun  33269  nodmon  33270  norn  33271  elno2  33274  noreson  33280