Theorem soex 7912

Description: If the relation in a strict order is a set, then the base field is also a set. (Contributed by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
soex	⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → 𝐴 ∈ V)

Proof of Theorem soex

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 486	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝐴 = ∅) → 𝐴 = ∅)
2		0ex 5308	. . 3 ⊢ ∅ ∈ V
3	1, 2	eqeltrdi 2842	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝐴 = ∅) → 𝐴 ∈ V)
4		n0 4347	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		vsnex 5430	. . . . . . . . 9 ⊢ {𝑥} ∈ V
6		dmexg 7894	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
7		rnexg 7895	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
8		unexg 7736	. . . . . . . . . 10 ⊢ ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
9	6, 7, 8	syl2anc 585	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
10		unexg 7736	. . . . . . . . 9 ⊢ (({𝑥} ∈ V ∧ (dom 𝑅 ∪ ran 𝑅) ∈ V) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
11	5, 9, 10	sylancr 588	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
12	11	ad2antlr 726	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
13		sossfld 6186	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
14	13	adantlr 714	. . . . . . . 8 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
15		ssundif 4488	. . . . . . . 8 ⊢ (𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ↔ (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
16	14, 15	sylibr 233	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)))
17	12, 16	ssexd 5325	. . . . . 6 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ V)
18	17	ex 414	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → (𝑥 ∈ 𝐴 → 𝐴 ∈ V))
19	18	exlimdv 1937	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ∈ V))
20	19	imp 408	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ ∃𝑥 𝑥 ∈ 𝐴) → 𝐴 ∈ V)
21	4, 20	sylan2b 595	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
22	3, 21	pm2.61dane 3030	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ≠ wne 2941  Vcvv 3475   ∖ cdif 3946   ∪ cun 3947   ⊆ wss 3949  ∅c0 4323  {csn 4629   Or wor 5588  dom cdm 5677  ran crn 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-po 5589  df-so 5590  df-cnv 5685  df-dm 5687  df-rn 5688

This theorem is referenced by:  ween  10030  zorn2lem1  10491  zorn2lem4  10494