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Theorem bnj1417 31235
Description: Technical lemma for bnj60 31256. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1417.1 (𝜑𝑅 FrSe 𝐴)
bnj1417.2 (𝜓 ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
bnj1417.3 (𝜒 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜓))
bnj1417.4 (𝜃 ↔ (𝜑𝑥𝐴𝜒))
bnj1417.5 𝐵 = ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
Assertion
Ref Expression
bnj1417 (𝜑 → ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦   𝜑,𝑥,𝑦   𝜓,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥,𝑦)   𝜃(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem bnj1417
StepHypRef Expression
1 bnj1417.1 . . . 4 (𝜑𝑅 FrSe 𝐴)
21biimpi 206 . . 3 (𝜑𝑅 FrSe 𝐴)
3 bnj1417.4 . . . . . 6 (𝜃 ↔ (𝜑𝑥𝐴𝜒))
4 bnj1418 31234 . . . . . . . . . . 11 (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥𝑅𝑥)
54adantl 481 . . . . . . . . . 10 ((𝜃𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑥𝑅𝑥)
63, 2bnj835 30955 . . . . . . . . . . . 12 (𝜃𝑅 FrSe 𝐴)
7 df-bnj15 30887 . . . . . . . . . . . . 13 (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Se 𝐴))
87simplbi 475 . . . . . . . . . . . 12 (𝑅 FrSe 𝐴𝑅 Fr 𝐴)
96, 8syl 17 . . . . . . . . . . 11 (𝜃𝑅 Fr 𝐴)
10 bnj213 31078 . . . . . . . . . . . 12 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐴
1110sseli 3632 . . . . . . . . . . 11 (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥𝐴)
12 frirr 5120 . . . . . . . . . . 11 ((𝑅 Fr 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
139, 11, 12syl2an 493 . . . . . . . . . 10 ((𝜃𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑥𝑅𝑥)
145, 13pm2.65da 599 . . . . . . . . 9 (𝜃 → ¬ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅))
15 nfv 1883 . . . . . . . . . . . . . 14 𝑦𝜑
16 nfv 1883 . . . . . . . . . . . . . 14 𝑦 𝑥𝐴
17 bnj1417.3 . . . . . . . . . . . . . . . 16 (𝜒 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜓))
1817bnj1095 30978 . . . . . . . . . . . . . . 15 (𝜒 → ∀𝑦𝜒)
1918nf5i 2064 . . . . . . . . . . . . . 14 𝑦𝜒
2015, 16, 19nf3an 1871 . . . . . . . . . . . . 13 𝑦(𝜑𝑥𝐴𝜒)
213, 20nfxfr 1819 . . . . . . . . . . . 12 𝑦𝜃
226ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
23 simplr 807 . . . . . . . . . . . . . . . . 17 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ pred(𝑥, 𝐴, 𝑅))
2410, 23sseldi 3634 . . . . . . . . . . . . . . . 16 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦𝐴)
25 simpr 476 . . . . . . . . . . . . . . . 16 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
26 bnj1125 31186 . . . . . . . . . . . . . . . 16 ((𝑅 FrSe 𝐴𝑦𝐴𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → trCl(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑦, 𝐴, 𝑅))
2722, 24, 25, 26syl3anc 1366 . . . . . . . . . . . . . . 15 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → trCl(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑦, 𝐴, 𝑅))
28 bnj1147 31188 . . . . . . . . . . . . . . . . . 18 trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
2928, 25sseldi 3634 . . . . . . . . . . . . . . . . 17 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥𝐴)
30 bnj906 31126 . . . . . . . . . . . . . . . . 17 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3122, 29, 30syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3231, 23sseldd 3637 . . . . . . . . . . . . . . 15 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅))
3327, 32sseldd 3637 . . . . . . . . . . . . . 14 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
3417biimpi 206 . . . . . . . . . . . . . . . . . 18 (𝜒 → ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜓))
353, 34bnj837 30957 . . . . . . . . . . . . . . . . 17 (𝜃 → ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜓))
3635ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜓))
37 bnj1418 31234 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
3837ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦𝑅𝑥)
39 rsp 2958 . . . . . . . . . . . . . . . 16 (∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜓) → (𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜓)))
4036, 24, 38, 39syl3c 66 . . . . . . . . . . . . . . 15 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → [𝑦 / 𝑥]𝜓)
41 vex 3234 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
42 bnj1417.2 . . . . . . . . . . . . . . . . 17 (𝜓 ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
43 eleq1 2718 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅)))
44 bnj1318 31219 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
4544eleq2d 2716 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝑦 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
4643, 45bitrd 268 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
4746notbid 307 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
4842, 47syl5bb 272 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝜓 ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
4941, 48sbcie 3503 . . . . . . . . . . . . . . 15 ([𝑦 / 𝑥]𝜓 ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
5040, 49sylib 208 . . . . . . . . . . . . . 14 (((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
5133, 50pm2.65da 599 . . . . . . . . . . . . 13 ((𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
5251ex 449 . . . . . . . . . . . 12 (𝜃 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)))
5321, 52ralrimi 2986 . . . . . . . . . . 11 (𝜃 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅) ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
54 ralnex 3021 . . . . . . . . . . 11 (∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅) ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅) ↔ ¬ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
5553, 54sylib 208 . . . . . . . . . 10 (𝜃 → ¬ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
56 eliun 4556 . . . . . . . . . 10 (𝑥 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
5755, 56sylnibr 318 . . . . . . . . 9 (𝜃 → ¬ 𝑥 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
58 ioran 510 . . . . . . . . 9 (¬ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ↔ (¬ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∧ ¬ 𝑥 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
5914, 57, 58sylanbrc 699 . . . . . . . 8 (𝜃 → ¬ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
603simp2bi 1097 . . . . . . . . . . 11 (𝜃𝑥𝐴)
61 bnj1417.5 . . . . . . . . . . . 12 𝐵 = ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
6261bnj1414 31231 . . . . . . . . . . 11 ((𝑅 FrSe 𝐴𝑥𝐴) → trCl(𝑥, 𝐴, 𝑅) = 𝐵)
636, 60, 62syl2anc 694 . . . . . . . . . 10 (𝜃 → trCl(𝑥, 𝐴, 𝑅) = 𝐵)
6463eleq2d 2716 . . . . . . . . 9 (𝜃 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑥𝐵))
6561bnj1138 30985 . . . . . . . . 9 (𝑥𝐵 ↔ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
6664, 65syl6bb 276 . . . . . . . 8 (𝜃 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))))
6759, 66mtbird 314 . . . . . . 7 (𝜃 → ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
6867, 42sylibr 224 . . . . . 6 (𝜃𝜓)
693, 68sylbir 225 . . . . 5 ((𝜑𝑥𝐴𝜒) → 𝜓)
70693exp 1283 . . . 4 (𝜑 → (𝑥𝐴 → (𝜒𝜓)))
7170ralrimiv 2994 . . 3 (𝜑 → ∀𝑥𝐴 (𝜒𝜓))
7217bnj1204 31206 . . 3 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜒𝜓)) → ∀𝑥𝐴 𝜓)
732, 71, 72syl2anc 694 . 2 (𝜑 → ∀𝑥𝐴 𝜓)
7442ralbii 3009 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
7573, 74sylib 208 1 (𝜑 → ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  [wsbc 3468  cun 3605  wss 3607   ciun 4552   class class class wbr 4685   Fr wfr 5099   predc-bnj14 30882   Se w-bnj13 30884   FrSe w-bnj15 30886   trClc-bnj18 30888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-bnj17 30881  df-bnj14 30883  df-bnj13 30885  df-bnj15 30887  df-bnj18 30889  df-bnj19 30891
This theorem is referenced by:  bnj1421  31236
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