Theorem soex 7768

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 485	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝐴 = ∅) → 𝐴 = ∅)
2		0ex 5231	. . 3 ⊢ ∅ ∈ V
3	1, 2	eqeltrdi 2847	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝐴 = ∅) → 𝐴 ∈ V)
4		n0 4280	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		snex 5354	. . . . . . . . 9 ⊢ {𝑥} ∈ V
6		dmexg 7750	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
7		rnexg 7751	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
8		unexg 7599	. . . . . . . . . 10 ⊢ ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
9	6, 7, 8	syl2anc 584	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
10		unexg 7599	. . . . . . . . 9 ⊢ (({𝑥} ∈ V ∧ (dom 𝑅 ∪ ran 𝑅) ∈ V) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
11	5, 9, 10	sylancr 587	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
12	11	ad2antlr 724	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
13		sossfld 6089	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
14	13	adantlr 712	. . . . . . . 8 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
15		ssundif 4418	. . . . . . . 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 5248	. . . . . 6 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ V)
18	17	ex 413	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → (𝑥 ∈ 𝐴 → 𝐴 ∈ V))
19	18	exlimdv 1936	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ∈ V))
20	19	imp 407	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ ∃𝑥 𝑥 ∈ 𝐴) → 𝐴 ∈ V)
21	4, 20	sylan2b 594	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
22	3, 21	pm2.61dane 3032	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  {csn 4561   Or wor 5502  dom cdm 5589  ran crn 5590

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-po 5503  df-so 5504  df-cnv 5597  df-dm 5599  df-rn 5600

This theorem is referenced by:  ween  9791  zorn2lem1  10252  zorn2lem4  10255