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Theorem soex 7388
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 479 . . 3 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝐴 = ∅) → 𝐴 = ∅)
2 0ex 5026 . . 3 ∅ ∈ V
31, 2syl6eqel 2866 . 2 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝐴 = ∅) → 𝐴 ∈ V)
4 n0 4158 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 snex 5140 . . . . . . . . 9 {𝑥} ∈ V
6 dmexg 7375 . . . . . . . . . 10 (𝑅𝑉 → dom 𝑅 ∈ V)
7 rnexg 7376 . . . . . . . . . 10 (𝑅𝑉 → ran 𝑅 ∈ V)
8 unexg 7236 . . . . . . . . . 10 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
96, 7, 8syl2anc 579 . . . . . . . . 9 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
10 unexg 7236 . . . . . . . . 9 (({𝑥} ∈ V ∧ (dom 𝑅 ∪ ran 𝑅) ∈ V) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
115, 9, 10sylancr 581 . . . . . . . 8 (𝑅𝑉 → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
1211ad2antlr 717 . . . . . . 7 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
13 sossfld 5834 . . . . . . . . 9 ((𝑅 Or 𝐴𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
1413adantlr 705 . . . . . . . 8 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
15 ssundif 4275 . . . . . . . 8 (𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)) ↔ (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
1614, 15sylibr 226 . . . . . . 7 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → 𝐴 ⊆ ({𝑥} ∪ (dom 𝑅 ∪ ran 𝑅)))
1712, 16ssexd 5042 . . . . . 6 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝑥𝐴) → 𝐴 ∈ V)
1817ex 403 . . . . 5 ((𝑅 Or 𝐴𝑅𝑉) → (𝑥𝐴𝐴 ∈ V))
1918exlimdv 1976 . . . 4 ((𝑅 Or 𝐴𝑅𝑉) → (∃𝑥 𝑥𝐴𝐴 ∈ V))
2019imp 397 . . 3 (((𝑅 Or 𝐴𝑅𝑉) ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ V)
214, 20sylan2b 587 . 2 (((𝑅 Or 𝐴𝑅𝑉) ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
223, 21pm2.61dane 3056 1 ((𝑅 Or 𝐴𝑅𝑉) → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wex 1823  wcel 2106  wne 2968  Vcvv 3397  cdif 3788  cun 3789  wss 3791  c0 4140  {csn 4397   Or wor 5273  dom cdm 5355  ran crn 5356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-po 5274  df-so 5275  df-cnv 5363  df-dm 5365  df-rn 5366
This theorem is referenced by:  ween  9191  zorn2lem1  9653  zorn2lem4  9656
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