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Theorem fin2solem 33842
Description: Lemma for fin2so 33843. (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 452 . . . . . . . . . 10 (((𝑦𝑥𝑧𝑥) ∧ 𝑤𝑥) ↔ (𝑤𝑥 ∧ (𝑦𝑥𝑧𝑥)))
2 3anass 1116 . . . . . . . . . 10 ((𝑤𝑥𝑦𝑥𝑧𝑥) ↔ (𝑤𝑥 ∧ (𝑦𝑥𝑧𝑥)))
31, 2bitr4i 269 . . . . . . . . 9 (((𝑦𝑥𝑧𝑥) ∧ 𝑤𝑥) ↔ (𝑤𝑥𝑦𝑥𝑧𝑥))
4 sotr 5222 . . . . . . . . 9 ((𝑅 Or 𝑥 ∧ (𝑤𝑥𝑦𝑥𝑧𝑥)) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
53, 4sylan2b 587 . . . . . . . 8 ((𝑅 Or 𝑥 ∧ ((𝑦𝑥𝑧𝑥) ∧ 𝑤𝑥)) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
65anassrs 459 . . . . . . 7 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑤𝑥) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
76ancomsd 457 . . . . . 6 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑤𝑥) → ((𝑦𝑅𝑧𝑤𝑅𝑦) → 𝑤𝑅𝑧))
87expdimp 444 . . . . 5 ((((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑤𝑥) ∧ 𝑦𝑅𝑧) → (𝑤𝑅𝑦𝑤𝑅𝑧))
98an32s 642 . . . 4 ((((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) ∧ 𝑤𝑥) → (𝑤𝑅𝑦𝑤𝑅𝑧))
109ss2rabdv 3845 . . 3 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧})
11 breq1 4814 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤𝑅𝑧𝑦𝑅𝑧))
1211elrab 3521 . . . . . . 7 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ↔ (𝑦𝑥𝑦𝑅𝑧))
1312biimpri 219 . . . . . 6 ((𝑦𝑥𝑦𝑅𝑧) → 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧})
1413adantll 705 . . . . 5 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧})
15 sonr 5221 . . . . . . 7 ((𝑅 Or 𝑥𝑦𝑥) → ¬ 𝑦𝑅𝑦)
16 breq1 4814 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤𝑅𝑦𝑦𝑅𝑦))
1716elrab 3521 . . . . . . . 8 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦} ↔ (𝑦𝑥𝑦𝑅𝑦))
1817simprbi 490 . . . . . . 7 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦} → 𝑦𝑅𝑦)
1915, 18nsyl 137 . . . . . 6 ((𝑅 Or 𝑥𝑦𝑥) → ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦})
2019adantr 472 . . . . 5 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦})
21 nelne1 3033 . . . . . 6 ((𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ∧ ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦}) → {𝑤𝑥𝑤𝑅𝑧} ≠ {𝑤𝑥𝑤𝑅𝑦})
2221necomd 2992 . . . . 5 ((𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ∧ ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦}) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
2314, 20, 22syl2anc 579 . . . 4 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
2423adantlrr 712 . . 3 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
25 vex 3353 . . . . . 6 𝑥 ∈ V
2625rabex 4975 . . . . 5 {𝑤𝑥𝑤𝑅𝑧} ∈ V
2726brrpss 7142 . . . 4 ({𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧} ↔ {𝑤𝑥𝑤𝑅𝑦} ⊊ {𝑤𝑥𝑤𝑅𝑧})
28 df-pss 3750 . . . 4 ({𝑤𝑥𝑤𝑅𝑦} ⊊ {𝑤𝑥𝑤𝑅𝑧} ↔ ({𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧} ∧ {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧}))
2927, 28bitri 266 . . 3 ({𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧} ↔ ({𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧} ∧ {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧}))
3010, 24, 29sylanbrc 578 . 2 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧})
3130ex 401 1 ((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) → (𝑦𝑅𝑧 → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1107  wcel 2155  wne 2937  {crab 3059  wss 3734  wpss 3735   class class class wbr 4811   Or wor 5199   [] crpss 7138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-po 5200  df-so 5201  df-xp 5285  df-rel 5286  df-rpss 7139
This theorem is referenced by:  fin2so  33843
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