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Theorem bnj1177 31352
Description: Technical lemma for bnj69 31356. 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 31039 . . . 4 (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Se 𝐴))
32simplbi 478 . . 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 31340 . . . . 5 trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
7 ssinss1 3972 . . . . 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 3764 . . 3 𝐶𝐴
109a1i 11 . 2 ((𝜑𝜓) → 𝐶𝐴)
11 bnj1177.17 . . . . . . 7 ((𝜑𝜓) → 𝑋𝐴)
12 bnj906 31278 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
131, 11, 12syl2anc 696 . . . . . 6 ((𝜑𝜓) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
14 ssrin 3969 . . . . . 6 ( pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) → ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
1513, 14syl 17 . . . . 5 ((𝜑𝜓) → ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
16 bnj1177.13 . . . . . . . 8 ((𝜑𝜓) → 𝐵𝐴)
17 bnj1177.2 . . . . . . . . . 10 (𝜓 ↔ (𝑋𝐵𝑦𝐵𝑦𝑅𝑋))
1817simp2bi 1138 . . . . . . . . 9 (𝜓𝑦𝐵)
1918adantl 473 . . . . . . . 8 ((𝜑𝜓) → 𝑦𝐵)
2016, 19sseldd 3733 . . . . . . 7 ((𝜑𝜓) → 𝑦𝐴)
2117simp3bi 1139 . . . . . . . 8 (𝜓𝑦𝑅𝑋)
2221adantl 473 . . . . . . 7 ((𝜑𝜓) → 𝑦𝑅𝑋)
23 bnj1152 31344 . . . . . . 7 (𝑦 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑦𝐴𝑦𝑅𝑋))
2420, 22, 23sylanbrc 701 . . . . . 6 ((𝜑𝜓) → 𝑦 ∈ pred(𝑋, 𝐴, 𝑅))
2524, 19elind 3929 . . . . 5 ((𝜑𝜓) → 𝑦 ∈ ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵))
2615, 25sseldd 3733 . . . 4 ((𝜑𝜓) → 𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
27 ne0i 4052 . . . 4 (𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
2826, 27syl 17 . . 3 ((𝜑𝜓) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
295neeq1i 2984 . . 3 (𝐶 ≠ ∅ ↔ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
3028, 29sylibr 224 . 2 ((𝜑𝜓) → 𝐶 ≠ ∅)
31 bnj893 31276 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
321, 11, 31syl2anc 696 . . 3 ((𝜑𝜓) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
33 inex1g 4941 . . . 4 ( trCl(𝑋, 𝐴, 𝑅) ∈ V → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ∈ V)
345, 33syl5eqel 2831 . . 3 ( trCl(𝑋, 𝐴, 𝑅) ∈ V → 𝐶 ∈ V)
3532, 34syl 17 . 2 ((𝜑𝜓) → 𝐶 ∈ V)
364, 10, 30, 35bnj951 31124 1 ((𝜑𝜓) → (𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1620  wcel 2127  wne 2920  Vcvv 3328  cin 3702  wss 3703  c0 4046   class class class wbr 4792   Fr wfr 5210  w-bnj17 31032   predc-bnj14 31034   Se w-bnj13 31036   FrSe w-bnj15 31038   trClc-bnj18 31040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-reg 8650  ax-inf2 8699
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-fal 1626  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-om 7219  df-1o 7717  df-bnj17 31033  df-bnj14 31035  df-bnj13 31037  df-bnj15 31039  df-bnj18 31041
This theorem is referenced by:  bnj1190  31354
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