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Theorem fin2solem 36053
Description: Lemma for fin2so 36054. (Contributed by Brendan Leahy, 29-Jun-2019.)
Assertion
Ref Expression
fin2solem ((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) → (𝑦𝑅𝑧 → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧}))
Distinct variable group:   𝑥,𝑤,𝑦,𝑧,𝑅

Proof of Theorem fin2solem
StepHypRef Expression
1 ancom 461 . . . . . . . . . 10 (((𝑦𝑥𝑧𝑥) ∧ 𝑤𝑥) ↔ (𝑤𝑥 ∧ (𝑦𝑥𝑧𝑥)))
2 3anass 1095 . . . . . . . . . 10 ((𝑤𝑥𝑦𝑥𝑧𝑥) ↔ (𝑤𝑥 ∧ (𝑦𝑥𝑧𝑥)))
31, 2bitr4i 277 . . . . . . . . 9 (((𝑦𝑥𝑧𝑥) ∧ 𝑤𝑥) ↔ (𝑤𝑥𝑦𝑥𝑧𝑥))
4 sotr 5568 . . . . . . . . 9 ((𝑅 Or 𝑥 ∧ (𝑤𝑥𝑦𝑥𝑧𝑥)) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
53, 4sylan2b 594 . . . . . . . 8 ((𝑅 Or 𝑥 ∧ ((𝑦𝑥𝑧𝑥) ∧ 𝑤𝑥)) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
65anassrs 468 . . . . . . 7 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑤𝑥) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
76ancomsd 466 . . . . . 6 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑤𝑥) → ((𝑦𝑅𝑧𝑤𝑅𝑦) → 𝑤𝑅𝑧))
87expdimp 453 . . . . 5 ((((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑤𝑥) ∧ 𝑦𝑅𝑧) → (𝑤𝑅𝑦𝑤𝑅𝑧))
98an32s 650 . . . 4 ((((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) ∧ 𝑤𝑥) → (𝑤𝑅𝑦𝑤𝑅𝑧))
109ss2rabdv 4032 . . 3 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧})
11 breq1 5107 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤𝑅𝑧𝑦𝑅𝑧))
1211elrab 3644 . . . . . . 7 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ↔ (𝑦𝑥𝑦𝑅𝑧))
1312biimpri 227 . . . . . 6 ((𝑦𝑥𝑦𝑅𝑧) → 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧})
1413adantll 712 . . . . 5 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧})
15 sonr 5567 . . . . . . 7 ((𝑅 Or 𝑥𝑦𝑥) → ¬ 𝑦𝑅𝑦)
16 breq1 5107 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤𝑅𝑦𝑦𝑅𝑦))
1716elrab 3644 . . . . . . . 8 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦} ↔ (𝑦𝑥𝑦𝑅𝑦))
1817simprbi 497 . . . . . . 7 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦} → 𝑦𝑅𝑦)
1915, 18nsyl 140 . . . . . 6 ((𝑅 Or 𝑥𝑦𝑥) → ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦})
2019adantr 481 . . . . 5 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦})
21 nelne1 3040 . . . . . 6 ((𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ∧ ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦}) → {𝑤𝑥𝑤𝑅𝑧} ≠ {𝑤𝑥𝑤𝑅𝑦})
2221necomd 2998 . . . . 5 ((𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ∧ ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦}) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
2314, 20, 22syl2anc 584 . . . 4 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
2423adantlrr 719 . . 3 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
25 vex 3448 . . . . . 6 𝑥 ∈ V
2625rabex 5288 . . . . 5 {𝑤𝑥𝑤𝑅𝑧} ∈ V
2726brrpss 7660 . . . 4 ({𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧} ↔ {𝑤𝑥𝑤𝑅𝑦} ⊊ {𝑤𝑥𝑤𝑅𝑧})
28 df-pss 3928 . . . 4 ({𝑤𝑥𝑤𝑅𝑦} ⊊ {𝑤𝑥𝑤𝑅𝑧} ↔ ({𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧} ∧ {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧}))
2927, 28bitri 274 . . 3 ({𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧} ↔ ({𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧} ∧ {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧}))
3010, 24, 29sylanbrc 583 . 2 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧})
3130ex 413 1 ((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) → (𝑦𝑅𝑧 → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087  wcel 2106  wne 2942  {crab 3406  wss 3909  wpss 3910   class class class wbr 5104   Or wor 5543   [] crpss 7656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-po 5544  df-so 5545  df-xp 5638  df-rel 5639  df-rpss 7657
This theorem is referenced by:  fin2so  36054
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