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Theorem soex 7059
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 477 . . 3 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝐴 = ∅) → 𝐴 = ∅)
2 0ex 4755 . . 3 ∅ ∈ V
31, 2syl6eqel 2712 . 2 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝐴 = ∅) → 𝐴 ∈ V)
4 n0 3912 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 snex 4874 . . . . . . . . 9 {𝑥} ∈ V
6 dmexg 7045 . . . . . . . . . 10 (𝑅𝑉 → dom 𝑅 ∈ V)
7 rnexg 7046 . . . . . . . . . 10 (𝑅𝑉 → ran 𝑅 ∈ V)
8 unexg 6913 . . . . . . . . . 10 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
96, 7, 8syl2anc 692 . . . . . . . . 9 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
10 unexg 6913 . . . . . . . . 9 (({𝑥} ∈ V ∧ (dom 𝑅 ∪ ran 𝑅) ∈ V) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
115, 9, 10sylancr 694 . . . . . . . 8 (𝑅𝑉 → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
1211ad2antlr 762 . . . . . . 7 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
13 sossfld 5543 . . . . . . . . 9 ((𝑅 Or 𝐴𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
1413adantlr 750 . . . . . . . 8 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
15 ssundif 4029 . . . . . . . 8 (𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ↔ (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
1614, 15sylibr 224 . . . . . . 7 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → 𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)))
1712, 16ssexd 4770 . . . . . 6 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → 𝐴 ∈ V)
1817ex 450 . . . . 5 ((𝑅 Or 𝐴𝑅𝑉) → (𝑥𝐴𝐴 ∈ V))
1918exlimdv 1863 . . . 4 ((𝑅 Or 𝐴𝑅𝑉) → (∃𝑥 𝑥𝐴𝐴 ∈ V))
2019imp 445 . . 3 (((𝑅 Or 𝐴𝑅𝑉) ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ V)
214, 20sylan2b 492 . 2 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
223, 21pm2.61dane 2883 1 ((𝑅 Or 𝐴𝑅𝑉) → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1992  wne 2796  Vcvv 3191  cdif 3557  cun 3558  wss 3560  c0 3896  {csn 4153   Or wor 4999  dom cdm 5079  ran crn 5080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-po 5000  df-so 5001  df-cnv 5087  df-dm 5089  df-rn 5090
This theorem is referenced by:  ween  8803  zorn2lem1  9263  zorn2lem4  9266
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