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Theorem bnj1177 34770
Description: Technical lemma for bnj69 34774. 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 34457 . . . 4 (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Se 𝐴))
32simplbi 496 . . 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 34758 . . . . 5 trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
7 ssinss1 4236 . . . . 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 34694 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
131, 11, 12syl2anc 582 . . . . . 6 ((𝜑𝜓) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
1413ssrind 4234 . . . . 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 480 . . . . . . . 8 ((𝜑𝜓) → 𝑦𝐵)
1915, 18sseldd 3977 . . . . . . 7 ((𝜑𝜓) → 𝑦𝐴)
2016simp3bi 1144 . . . . . . . 8 (𝜓𝑦𝑅𝑋)
2120adantl 480 . . . . . . 7 ((𝜑𝜓) → 𝑦𝑅𝑋)
22 bnj1152 34762 . . . . . . 7 (𝑦 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑦𝐴𝑦𝑅𝑋))
2319, 21, 22sylanbrc 581 . . . . . 6 ((𝜑𝜓) → 𝑦 ∈ pred(𝑋, 𝐴, 𝑅))
2423, 18elind 4192 . . . . 5 ((𝜑𝜓) → 𝑦 ∈ ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵))
2514, 24sseldd 3977 . . . 4 ((𝜑𝜓) → 𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
2625ne0d 4335 . . 3 ((𝜑𝜓) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
275neeq1i 2994 . . 3 (𝐶 ≠ ∅ ↔ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
2826, 27sylibr 233 . 2 ((𝜑𝜓) → 𝐶 ≠ ∅)
29 bnj893 34692 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
301, 11, 29syl2anc 582 . . 3 ((𝜑𝜓) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
31 inex1g 5320 . . . 4 ( trCl(𝑋, 𝐴, 𝑅) ∈ V → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ∈ V)
325, 31eqeltrid 2829 . . 3 ( trCl(𝑋, 𝐴, 𝑅) ∈ V → 𝐶 ∈ V)
3330, 32syl 17 . 2 ((𝜑𝜓) → 𝐶 ∈ V)
344, 10, 28, 33bnj951 34539 1 ((𝜑𝜓) → (𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  Vcvv 3461  cin 3943  wss 3944  c0 4322   class class class wbr 5149   Fr wfr 5630  w-bnj17 34450   predc-bnj14 34452   Se w-bnj13 34454   FrSe w-bnj15 34456   trClc-bnj18 34458
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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-reg 9622  ax-inf2 9671
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-1o 8487  df-bnj17 34451  df-bnj14 34453  df-bnj13 34455  df-bnj15 34457  df-bnj18 34459
This theorem is referenced by:  bnj1190  34772
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