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Theorem soex 7859
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 𝐴𝑅𝑉) → 𝐴 ∈ V)

Proof of Theorem soex
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 486 . . 3 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝐴 = ∅) → 𝐴 = ∅)
2 0ex 5265 . . 3 ∅ ∈ V
31, 2eqeltrdi 2842 . 2 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝐴 = ∅) → 𝐴 ∈ V)
4 n0 4307 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 vsnex 5387 . . . . . . . . 9 {𝑥} ∈ V
6 dmexg 7841 . . . . . . . . . 10 (𝑅𝑉 → dom 𝑅 ∈ V)
7 rnexg 7842 . . . . . . . . . 10 (𝑅𝑉 → ran 𝑅 ∈ V)
8 unexg 7684 . . . . . . . . . 10 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
96, 7, 8syl2anc 585 . . . . . . . . 9 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
10 unexg 7684 . . . . . . . . 9 (({𝑥} ∈ V ∧ (dom 𝑅 ∪ ran 𝑅) ∈ V) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
115, 9, 10sylancr 588 . . . . . . . 8 (𝑅𝑉 → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
1211ad2antlr 726 . . . . . . 7 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
13 sossfld 6139 . . . . . . . . 9 ((𝑅 Or 𝐴𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
1413adantlr 714 . . . . . . . 8 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
15 ssundif 4446 . . . . . . . 8 (𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ↔ (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
1614, 15sylibr 233 . . . . . . 7 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → 𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)))
1712, 16ssexd 5282 . . . . . 6 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → 𝐴 ∈ V)
1817ex 414 . . . . 5 ((𝑅 Or 𝐴𝑅𝑉) → (𝑥𝐴𝐴 ∈ V))
1918exlimdv 1937 . . . 4 ((𝑅 Or 𝐴𝑅𝑉) → (∃𝑥 𝑥𝐴𝐴 ∈ V))
2019imp 408 . . 3 (((𝑅 Or 𝐴𝑅𝑉) ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ V)
214, 20sylan2b 595 . 2 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
223, 21pm2.61dane 3029 1 ((𝑅 Or 𝐴𝑅𝑉) → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2940  Vcvv 3444  cdif 3908  cun 3909  wss 3911  c0 4283  {csn 4587   Or wor 5545  dom cdm 5634  ran crn 5635
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 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-po 5546  df-so 5547  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  ween  9976  zorn2lem1  10437  zorn2lem4  10440
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