Theorem elnoOLD 27673

Description: Obsolete version of elno 27672 as of 5-Jun-2025. (Contributed by Scott Fenton, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elnoOLD	⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})

Distinct variable group:   𝑥,𝐴

Proof of Theorem elnoOLD

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3482	. 2 ⊢ (𝐴 ∈ No → 𝐴 ∈ V)
2		fex 7235	. . . 4 ⊢ ((𝐴:𝑥⟶{1o, 2o} ∧ 𝑥 ∈ On) → 𝐴 ∈ V)
3	2	ancoms 457	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) → 𝐴 ∈ V)
4	3	rexlimiva 3137	. 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V)
5		feq1 6701	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝐴:𝑥⟶{1o, 2o}))
6	5	rexbidv 3169	. . 3 ⊢ (𝑓 = 𝐴 → (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
7		df-no 27669	. . 3 ⊢ No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}}
8	6, 7	elab2g 3667	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}))
9	1, 4, 8	pm5.21nii 377	1 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})

Syntax hints:   ↔ wb 205   = wceq 1534   ∈ wcel 2099  ∃wrex 3060  Vcvv 3462  {cpr 4625  Oncon0 6368  ⟶wf 6542  1_oc1o 8481  2_oc2o 8482   No csur 27666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-no 27669

This theorem is referenced by: (None)