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Theorem fin2solem 33729
Description: Lemma for fin2so 33730. (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 1080 . . . . . . . . . 10 ((𝑤𝑥𝑦𝑥𝑧𝑥) ↔ (𝑤𝑥 ∧ (𝑦𝑥𝑧𝑥)))
31, 2bitr4i 267 . . . . . . . . 9 (((𝑦𝑥𝑧𝑥) ∧ 𝑤𝑥) ↔ (𝑤𝑥𝑦𝑥𝑧𝑥))
4 sotr 5193 . . . . . . . . 9 ((𝑅 Or 𝑥 ∧ (𝑤𝑥𝑦𝑥𝑧𝑥)) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
53, 4sylan2b 575 . . . . . . . 8 ((𝑅 Or 𝑥 ∧ ((𝑦𝑥𝑧𝑥) ∧ 𝑤𝑥)) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
65anassrs 458 . . . . . . 7 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑤𝑥) → ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
76ancomsd 456 . . . . . 6 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑤𝑥) → ((𝑦𝑅𝑧𝑤𝑅𝑦) → 𝑤𝑅𝑧))
87expdimp 440 . . . . 5 ((((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑤𝑥) ∧ 𝑦𝑅𝑧) → (𝑤𝑅𝑦𝑤𝑅𝑧))
98an32s 625 . . . 4 ((((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) ∧ 𝑤𝑥) → (𝑤𝑅𝑦𝑤𝑅𝑧))
109ss2rabdv 3833 . . 3 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧})
11 breq1 4790 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤𝑅𝑧𝑦𝑅𝑧))
1211elrab 3516 . . . . . . 7 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ↔ (𝑦𝑥𝑦𝑅𝑧))
1312biimpri 218 . . . . . 6 ((𝑦𝑥𝑦𝑅𝑧) → 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧})
1413adantll 687 . . . . 5 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧})
15 sonr 5192 . . . . . . 7 ((𝑅 Or 𝑥𝑦𝑥) → ¬ 𝑦𝑅𝑦)
16 breq1 4790 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤𝑅𝑦𝑦𝑅𝑦))
1716elrab 3516 . . . . . . . 8 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦} ↔ (𝑦𝑥𝑦𝑅𝑦))
1817simprbi 480 . . . . . . 7 (𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦} → 𝑦𝑅𝑦)
1915, 18nsyl 137 . . . . . 6 ((𝑅 Or 𝑥𝑦𝑥) → ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦})
2019adantr 466 . . . . 5 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦})
21 nelne1 3039 . . . . . 6 ((𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ∧ ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦}) → {𝑤𝑥𝑤𝑅𝑧} ≠ {𝑤𝑥𝑤𝑅𝑦})
2221necomd 2998 . . . . 5 ((𝑦 ∈ {𝑤𝑥𝑤𝑅𝑧} ∧ ¬ 𝑦 ∈ {𝑤𝑥𝑤𝑅𝑦}) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
2314, 20, 22syl2anc 567 . . . 4 (((𝑅 Or 𝑥𝑦𝑥) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
2423adantlrr 694 . . 3 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧})
25 vex 3354 . . . . . 6 𝑥 ∈ V
2625rabex 4947 . . . . 5 {𝑤𝑥𝑤𝑅𝑧} ∈ V
2726brrpss 7088 . . . 4 ({𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧} ↔ {𝑤𝑥𝑤𝑅𝑦} ⊊ {𝑤𝑥𝑤𝑅𝑧})
28 df-pss 3740 . . . 4 ({𝑤𝑥𝑤𝑅𝑦} ⊊ {𝑤𝑥𝑤𝑅𝑧} ↔ ({𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧} ∧ {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧}))
2927, 28bitri 264 . . 3 ({𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧} ↔ ({𝑤𝑥𝑤𝑅𝑦} ⊆ {𝑤𝑥𝑤𝑅𝑧} ∧ {𝑤𝑥𝑤𝑅𝑦} ≠ {𝑤𝑥𝑤𝑅𝑧}))
3010, 24, 29sylanbrc 566 . 2 (((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧})
3130ex 397 1 ((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) → (𝑦𝑅𝑧 → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1071  wcel 2145  wne 2943  {crab 3065  wss 3724  wpss 3725   class class class wbr 4787   Or wor 5170   [] crpss 7084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-po 5171  df-so 5172  df-xp 5256  df-rel 5257  df-rpss 7085
This theorem is referenced by:  fin2so  33730
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