Theorem soex 7873

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 484	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝐴 = ∅) → 𝐴 = ∅)
2		0ex 5254	. . 3 ⊢ ∅ ∈ V
3	1, 2	eqeltrdi 2845	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝐴 = ∅) → 𝐴 ∈ V)
4		n0 4307	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		vsnex 5381	. . . . . . . . 9 ⊢ {𝑥} ∈ V
6		dmexg 7853	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
7		rnexg 7854	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
8		unexg 7698	. . . . . . . . . 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 7698	. . . . . . . . 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 728	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
13		sossfld 6152	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
14	13	adantlr 716	. . . . . . . 8 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
15		ssundif 4442	. . . . . . . 8 ⊢ (𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ↔ (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
16	14, 15	sylibr 234	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)))
17	12, 16	ssexd 5271	. . . . . 6 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ V)
18	17	ex 412	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → (𝑥 ∈ 𝐴 → 𝐴 ∈ V))
19	18	exlimdv 1935	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ∈ V))
20	19	imp 406	. . 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 3020	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  ∅c0 4287  {csn 4582   Or wor 5539  dom cdm 5632  ran crn 5633

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-po 5540  df-so 5541  df-cnv 5640  df-dm 5642  df-rn 5643

This theorem is referenced by:  ween  9957  zorn2lem1  10418  zorn2lem4  10421