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Theorem bnj1177 32296
Description: Technical lemma for bnj69 32300. 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.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1177.2 (𝜓 ↔ (𝑋𝐵𝑦𝐵𝑦𝑅𝑋))
bnj1177.3 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
bnj1177.9 ((𝜑𝜓) → 𝑅 FrSe 𝐴)
bnj1177.13 ((𝜑𝜓) → 𝐵𝐴)
bnj1177.17 ((𝜑𝜓) → 𝑋𝐴)
Assertion
Ref Expression
bnj1177 ((𝜑𝜓) → (𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V))

Proof of Theorem bnj1177
StepHypRef Expression
1 bnj1177.9 . . 3 ((𝜑𝜓) → 𝑅 FrSe 𝐴)
2 df-bnj15 31981 . . . 4 (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Se 𝐴))
32simplbi 501 . . 3 (𝑅 FrSe 𝐴𝑅 Fr 𝐴)
41, 3syl 17 . 2 ((𝜑𝜓) → 𝑅 Fr 𝐴)
5 bnj1177.3 . . . 4 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
6 bnj1147 32284 . . . . 5 trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
7 ssinss1 4197 . . . . 5 ( trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴 → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴)
86, 7ax-mp 5 . . . 4 ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴
95, 8eqsstri 3985 . . 3 𝐶𝐴
109a1i 11 . 2 ((𝜑𝜓) → 𝐶𝐴)
11 bnj1177.17 . . . . . . 7 ((𝜑𝜓) → 𝑋𝐴)
12 bnj906 32220 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
131, 11, 12syl2anc 587 . . . . . 6 ((𝜑𝜓) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
1413ssrind 4195 . . . . 5 ((𝜑𝜓) → ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
15 bnj1177.13 . . . . . . . 8 ((𝜑𝜓) → 𝐵𝐴)
16 bnj1177.2 . . . . . . . . . 10 (𝜓 ↔ (𝑋𝐵𝑦𝐵𝑦𝑅𝑋))
1716simp2bi 1143 . . . . . . . . 9 (𝜓𝑦𝐵)
1817adantl 485 . . . . . . . 8 ((𝜑𝜓) → 𝑦𝐵)
1915, 18sseldd 3952 . . . . . . 7 ((𝜑𝜓) → 𝑦𝐴)
2016simp3bi 1144 . . . . . . . 8 (𝜓𝑦𝑅𝑋)
2120adantl 485 . . . . . . 7 ((𝜑𝜓) → 𝑦𝑅𝑋)
22 bnj1152 32288 . . . . . . 7 (𝑦 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑦𝐴𝑦𝑅𝑋))
2319, 21, 22sylanbrc 586 . . . . . 6 ((𝜑𝜓) → 𝑦 ∈ pred(𝑋, 𝐴, 𝑅))
2423, 18elind 4154 . . . . 5 ((𝜑𝜓) → 𝑦 ∈ ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵))
2514, 24sseldd 3952 . . . 4 ((𝜑𝜓) → 𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
2625ne0d 4282 . . 3 ((𝜑𝜓) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
275neeq1i 3077 . . 3 (𝐶 ≠ ∅ ↔ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
2826, 27sylibr 237 . 2 ((𝜑𝜓) → 𝐶 ≠ ∅)
29 bnj893 32218 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
301, 11, 29syl2anc 587 . . 3 ((𝜑𝜓) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
31 inex1g 5204 . . . 4 ( trCl(𝑋, 𝐴, 𝑅) ∈ V → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ∈ V)
325, 31eqeltrid 2920 . . 3 ( trCl(𝑋, 𝐴, 𝑅) ∈ V → 𝐶 ∈ V)
3330, 32syl 17 . 2 ((𝜑𝜓) → 𝐶 ∈ V)
344, 10, 28, 33bnj951 32065 1 ((𝜑𝜓) → (𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3013  Vcvv 3479  cin 3917  wss 3918  c0 4274   class class class wbr 5047   Fr wfr 5492  w-bnj17 31974   predc-bnj14 31976   Se w-bnj13 31978   FrSe w-bnj15 31980   trClc-bnj18 31982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-reg 9040  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7564  df-1o 8085  df-bnj17 31975  df-bnj14 31977  df-bnj13 31979  df-bnj15 31981  df-bnj18 31983
This theorem is referenced by:  bnj1190  32298
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