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Theorem fin2solem 37593
Description: Lemma for fin2so 37594. (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 460 . . . . . . . . . 10 (((𝑦𝑥𝑧𝑥) ∧ 𝑤𝑥) ↔ (𝑤𝑥 ∧ (𝑦𝑥𝑧𝑥)))
2 3anass 1094 . . . . . . . . . 10 ((𝑤𝑥𝑦𝑥𝑧𝑥) ↔ (𝑤𝑥 ∧ (𝑦𝑥𝑧𝑥)))
31, 2bitr4i 278 . . . . . . . . 9 (((𝑦𝑥𝑧𝑥) ∧ 𝑤𝑥) ↔ (𝑤𝑥𝑦𝑥𝑧𝑥))
4 sotr 5622 . . . . . . . . 9 ((𝑅 Or 𝑥 ∧ (𝑤𝑥𝑦𝑥𝑧𝑥)) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
53, 4sylan2b 594 . . . . . . . 8 ((𝑅 Or 𝑥 ∧ ((𝑦𝑥𝑧𝑥) ∧ 𝑤𝑥)) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
65anassrs 467 . . . . . . 7 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑤𝑥) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
76ancomsd 465 . . . . . 6 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑤𝑥) → ((𝑦𝑅𝑧𝑤𝑅𝑦) → 𝑤𝑅𝑧))
87expdimp 452 . . . . 5 ((((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑤𝑥) ∧ 𝑦𝑅𝑧) → (𝑤𝑅𝑦𝑤𝑅𝑧))
98an32s 652 . . . 4 ((((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) ∧ 𝑤𝑥) → (𝑤𝑅𝑦𝑤𝑅𝑧))
109ss2rabdv 4086 . . 3 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧})
11 breq1 5151 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤𝑅𝑧𝑦𝑅𝑧))
1211elrab 3695 . . . . . . 7 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ↔ (𝑦𝑥𝑦𝑅𝑧))
1312biimpri 228 . . . . . 6 ((𝑦𝑥𝑦𝑅𝑧) → 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧})
1413adantll 714 . . . . 5 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧})
15 sonr 5621 . . . . . . 7 ((𝑅 Or 𝑥𝑦𝑥) → ¬ 𝑦𝑅𝑦)
16 breq1 5151 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤𝑅𝑦𝑦𝑅𝑦))
1716elrab 3695 . . . . . . . 8 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦} ↔ (𝑦𝑥𝑦𝑅𝑦))
1817simprbi 496 . . . . . . 7 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦} → 𝑦𝑅𝑦)
1915, 18nsyl 140 . . . . . 6 ((𝑅 Or 𝑥𝑦𝑥) → ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦})
2019adantr 480 . . . . 5 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦})
21 nelne1 3037 . . . . . 6 ((𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ∧ ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦}) → {𝑤𝑥𝑤𝑅𝑧} ≠ {𝑤𝑥𝑤𝑅𝑦})
2221necomd 2994 . . . . 5 ((𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ∧ ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦}) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
2314, 20, 22syl2anc 584 . . . 4 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
2423adantlrr 721 . . 3 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
25 vex 3482 . . . . . 6 𝑥 ∈ V
2625rabex 5345 . . . . 5 {𝑤𝑥𝑤𝑅𝑧} ∈ V
2726brrpss 7745 . . . 4 ({𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧} ↔ {𝑤𝑥𝑤𝑅𝑦} ⊊ {𝑤𝑥𝑤𝑅𝑧})
28 df-pss 3983 . . . 4 ({𝑤𝑥𝑤𝑅𝑦} ⊊ {𝑤𝑥𝑤𝑅𝑧} ↔ ({𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧} ∧ {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧}))
2927, 28bitri 275 . . 3 ({𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧} ↔ ({𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧} ∧ {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧}))
3010, 24, 29sylanbrc 583 . 2 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧})
3130ex 412 1 ((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) → (𝑦𝑅𝑧 → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086  wcel 2106  wne 2938  {crab 3433  wss 3963  wpss 3964   class class class wbr 5148   Or wor 5596   [] crpss 7741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-rpss 7742
This theorem is referenced by:  fin2so  37594
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