Theorem soex 7933

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 490	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝐴 = ∅) → 𝐴 = ∅)
2		0ex 5260	. . 3 ⊢ ∅ ∈ V
3	1, 2	eqeltrdi 2868	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝐴 = ∅) → 𝐴 ∈ V)
4		n0 4299	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		vsnex 5392	. . . . . . . . 9 ⊢ {𝑥} ∈ V
6		dmexg 7913	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
7		rnexg 7914	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
8		unexg 7760	. . . . . . . . . 10 ⊢ ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
9	6, 7, 8	syl2anc 596	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
10		unexg 7760	. . . . . . . . 9 ⊢ (({𝑥} ∈ V ∧ (dom 𝑅 ∪ ran 𝑅) ∈ V) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
11	5, 9, 10	sylancr 599	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
12	11	ad2antlr 740	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
13		sossfld 6177	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
14	13	adantlr 728	. . . . . . . 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 237	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)))
17	12, 16	ssexd 5285	. . . . . 6 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ V)
18	17	ex 418	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → (𝑥 ∈ 𝐴 → 𝐴 ∈ V))
19	18	exlimdv 1965	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ∈ V))
20	19	imp 412	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ ∃𝑥 𝑥 ∈ 𝐴) → 𝐴 ∈ V)
21	4, 20	sylan2b 606	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
22	3, 21	pm2.61dane 3042	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450   ⊆ wss 3895   ∖ cdif 4045   ∪ cun 4149  ∅c0 4278  {csn 4583   Or wor 5557  dom cdm 5650  ran crn 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-po 5558  df-so 5559  df-cnv 5658  df-dm 5660  df-rn 5661

This theorem is used by:  ween  10115  zorn2lem1  10575  zorn2lem4  10578