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Theorem fin2solem 36778
Description: Lemma for fin2so 36779. (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 277 . . . . . . . . 9 (((𝑦𝑥𝑧𝑥) ∧ 𝑤𝑥) ↔ (𝑤𝑥𝑦𝑥𝑧𝑥))
4 sotr 5613 . . . . . . . . 9 ((𝑅 Or 𝑥 ∧ (𝑤𝑥𝑦𝑥𝑧𝑥)) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
53, 4sylan2b 593 . . . . . . . 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 649 . . . 4 ((((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) ∧ 𝑤𝑥) → (𝑤𝑅𝑦𝑤𝑅𝑧))
109ss2rabdv 4074 . . 3 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧})
11 breq1 5152 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤𝑅𝑧𝑦𝑅𝑧))
1211elrab 3684 . . . . . . 7 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ↔ (𝑦𝑥𝑦𝑅𝑧))
1312biimpri 227 . . . . . 6 ((𝑦𝑥𝑦𝑅𝑧) → 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧})
1413adantll 711 . . . . 5 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧})
15 sonr 5612 . . . . . . 7 ((𝑅 Or 𝑥𝑦𝑥) → ¬ 𝑦𝑅𝑦)
16 breq1 5152 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤𝑅𝑦𝑦𝑅𝑦))
1716elrab 3684 . . . . . . . 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 3038 . . . . . 6 ((𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ∧ ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦}) → {𝑤𝑥𝑤𝑅𝑧} ≠ {𝑤𝑥𝑤𝑅𝑦})
2221necomd 2995 . . . . 5 ((𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ∧ ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦}) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
2314, 20, 22syl2anc 583 . . . 4 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
2423adantlrr 718 . . 3 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
25 vex 3477 . . . . . 6 𝑥 ∈ V
2625rabex 5333 . . . . 5 {𝑤𝑥𝑤𝑅𝑧} ∈ V
2726brrpss 7719 . . . 4 ({𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧} ↔ {𝑤𝑥𝑤𝑅𝑦} ⊊ {𝑤𝑥𝑤𝑅𝑧})
28 df-pss 3968 . . . 4 ({𝑤𝑥𝑤𝑅𝑦} ⊊ {𝑤𝑥𝑤𝑅𝑧} ↔ ({𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧} ∧ {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧}))
2927, 28bitri 274 . . 3 ({𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧} ↔ ({𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧} ∧ {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧}))
3010, 24, 29sylanbrc 582 . 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 2105  wne 2939  {crab 3431  wss 3949  wpss 3950   class class class wbr 5149   Or wor 5588   [] crpss 7715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-rpss 7716
This theorem is referenced by:  fin2so  36779
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