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Theorem soex 7615
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 485 . . 3 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝐴 = ∅) → 𝐴 = ∅)
2 0ex 5202 . . 3 ∅ ∈ V
31, 2syl6eqel 2918 . 2 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝐴 = ∅) → 𝐴 ∈ V)
4 n0 4307 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 snex 5322 . . . . . . . . 9 {𝑥} ∈ V
6 dmexg 7602 . . . . . . . . . 10 (𝑅𝑉 → dom 𝑅 ∈ V)
7 rnexg 7603 . . . . . . . . . 10 (𝑅𝑉 → ran 𝑅 ∈ V)
8 unexg 7461 . . . . . . . . . 10 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
96, 7, 8syl2anc 584 . . . . . . . . 9 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
10 unexg 7461 . . . . . . . . 9 (({𝑥} ∈ V ∧ (dom 𝑅 ∪ ran 𝑅) ∈ V) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
115, 9, 10sylancr 587 . . . . . . . 8 (𝑅𝑉 → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
1211ad2antlr 723 . . . . . . 7 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
13 sossfld 6036 . . . . . . . . 9 ((𝑅 Or 𝐴𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
1413adantlr 711 . . . . . . . 8 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
15 ssundif 4429 . . . . . . . 8 (𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ↔ (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
1614, 15sylibr 235 . . . . . . 7 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → 𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)))
1712, 16ssexd 5219 . . . . . 6 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → 𝐴 ∈ V)
1817ex 413 . . . . 5 ((𝑅 Or 𝐴𝑅𝑉) → (𝑥𝐴𝐴 ∈ V))
1918exlimdv 1925 . . . 4 ((𝑅 Or 𝐴𝑅𝑉) → (∃𝑥 𝑥𝐴𝐴 ∈ V))
2019imp 407 . . 3 (((𝑅 Or 𝐴𝑅𝑉) ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ V)
214, 20sylan2b 593 . 2 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
223, 21pm2.61dane 3101 1 ((𝑅 Or 𝐴𝑅𝑉) → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  wne 3013  Vcvv 3492  cdif 3930  cun 3931  wss 3933  c0 4288  {csn 4557   Or wor 5466  dom cdm 5548  ran crn 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-po 5467  df-so 5468  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by:  ween  9449  zorn2lem1  9906  zorn2lem4  9909
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