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Theorem bnj1177 34546
Description: Technical lemma for bnj69 34550. 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 34233 . . . 4 (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Se 𝐴))
32simplbi 497 . . 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 34534 . . . . 5 trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
7 ssinss1 4232 . . . . 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 4011 . . 3 𝐶𝐴
109a1i 11 . 2 ((𝜑𝜓) → 𝐶𝐴)
11 bnj1177.17 . . . . . . 7 ((𝜑𝜓) → 𝑋𝐴)
12 bnj906 34470 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
131, 11, 12syl2anc 583 . . . . . 6 ((𝜑𝜓) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
1413ssrind 4230 . . . . 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 481 . . . . . . . 8 ((𝜑𝜓) → 𝑦𝐵)
1915, 18sseldd 3978 . . . . . . 7 ((𝜑𝜓) → 𝑦𝐴)
2016simp3bi 1144 . . . . . . . 8 (𝜓𝑦𝑅𝑋)
2120adantl 481 . . . . . . 7 ((𝜑𝜓) → 𝑦𝑅𝑋)
22 bnj1152 34538 . . . . . . 7 (𝑦 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑦𝐴𝑦𝑅𝑋))
2319, 21, 22sylanbrc 582 . . . . . 6 ((𝜑𝜓) → 𝑦 ∈ pred(𝑋, 𝐴, 𝑅))
2423, 18elind 4189 . . . . 5 ((𝜑𝜓) → 𝑦 ∈ ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵))
2514, 24sseldd 3978 . . . 4 ((𝜑𝜓) → 𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
2625ne0d 4330 . . 3 ((𝜑𝜓) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
275neeq1i 2999 . . 3 (𝐶 ≠ ∅ ↔ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
2826, 27sylibr 233 . 2 ((𝜑𝜓) → 𝐶 ≠ ∅)
29 bnj893 34468 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
301, 11, 29syl2anc 583 . . 3 ((𝜑𝜓) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
31 inex1g 5312 . . . 4 ( trCl(𝑋, 𝐴, 𝑅) ∈ V → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ∈ V)
325, 31eqeltrid 2831 . . 3 ( trCl(𝑋, 𝐴, 𝑅) ∈ V → 𝐶 ∈ V)
3330, 32syl 17 . 2 ((𝜑𝜓) → 𝐶 ∈ V)
344, 10, 28, 33bnj951 34315 1 ((𝜑𝜓) → (𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2934  Vcvv 3468  cin 3942  wss 3943  c0 4317   class class class wbr 5141   Fr wfr 5621  w-bnj17 34226   predc-bnj14 34228   Se w-bnj13 34230   FrSe w-bnj15 34232   trClc-bnj18 34234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-reg 9589  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7853  df-1o 8467  df-bnj17 34227  df-bnj14 34229  df-bnj13 34231  df-bnj15 34233  df-bnj18 34235
This theorem is referenced by:  bnj1190  34548
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