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Theorem bnj1177 33675
Description: Technical lemma for bnj69 33679. 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 33362 . . . 4 (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Se 𝐴))
32simplbi 499 . . 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 33663 . . . . 5 trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
7 ssinss1 4198 . . . . 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 3979 . . 3 𝐶𝐴
109a1i 11 . 2 ((𝜑𝜓) → 𝐶𝐴)
11 bnj1177.17 . . . . . . 7 ((𝜑𝜓) → 𝑋𝐴)
12 bnj906 33599 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
131, 11, 12syl2anc 585 . . . . . 6 ((𝜑𝜓) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
1413ssrind 4196 . . . . 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 1147 . . . . . . . . 9 (𝜓𝑦𝐵)
1817adantl 483 . . . . . . . 8 ((𝜑𝜓) → 𝑦𝐵)
1915, 18sseldd 3946 . . . . . . 7 ((𝜑𝜓) → 𝑦𝐴)
2016simp3bi 1148 . . . . . . . 8 (𝜓𝑦𝑅𝑋)
2120adantl 483 . . . . . . 7 ((𝜑𝜓) → 𝑦𝑅𝑋)
22 bnj1152 33667 . . . . . . 7 (𝑦 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑦𝐴𝑦𝑅𝑋))
2319, 21, 22sylanbrc 584 . . . . . 6 ((𝜑𝜓) → 𝑦 ∈ pred(𝑋, 𝐴, 𝑅))
2423, 18elind 4155 . . . . 5 ((𝜑𝜓) → 𝑦 ∈ ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵))
2514, 24sseldd 3946 . . . 4 ((𝜑𝜓) → 𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
2625ne0d 4296 . . 3 ((𝜑𝜓) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
275neeq1i 3005 . . 3 (𝐶 ≠ ∅ ↔ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
2826, 27sylibr 233 . 2 ((𝜑𝜓) → 𝐶 ≠ ∅)
29 bnj893 33597 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
301, 11, 29syl2anc 585 . . 3 ((𝜑𝜓) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
31 inex1g 5277 . . . 4 ( trCl(𝑋, 𝐴, 𝑅) ∈ V → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ∈ V)
325, 31eqeltrid 2838 . . 3 ( trCl(𝑋, 𝐴, 𝑅) ∈ V → 𝐶 ∈ V)
3330, 32syl 17 . 2 ((𝜑𝜓) → 𝐶 ∈ V)
344, 10, 28, 33bnj951 33444 1 ((𝜑𝜓) → (𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  Vcvv 3444  cin 3910  wss 3911  c0 4283   class class class wbr 5106   Fr wfr 5586  w-bnj17 33355   predc-bnj14 33357   Se w-bnj13 33359   FrSe w-bnj15 33361   trClc-bnj18 33363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-reg 9533  ax-inf2 9582
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804  df-1o 8413  df-bnj17 33356  df-bnj14 33358  df-bnj13 33360  df-bnj15 33362  df-bnj18 33364
This theorem is referenced by:  bnj1190  33677
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