Theorem bnj1190 32988

Description: Technical lemma for bnj69 32990. 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
bnj1190.1	⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅))
bnj1190.2	⊢ (𝜓 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑥))

Assertion

Ref	Expression
bnj1190	⊢ ((𝜑 ∧ 𝜓) → ∃𝑤 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑤)

Distinct variable groups:   𝑤,𝐵,𝑥,𝑧   𝑦,𝐵,𝑥,𝑧   𝑤,𝑅,𝑥,𝑧   𝑦,𝑅

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)   𝐴(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem bnj1190

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1190.1	. . . . . . 7 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅))
2	1	simp2bi 1145	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
3	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴)
4		eqid 2738	. . . . . 6 ⊢ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) = ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)
5		bnj1190.2	. . . . . . . . 9 ⊢ (𝜓 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑥))
6	1	simp1bi 1144	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 FrSe 𝐴)
7	6	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 FrSe 𝐴)
8	5	simp1bi 1144	. . . . . . . . . 10 ⊢ (𝜓 → 𝑥 ∈ 𝐵)
9		ssel2 3916	. . . . . . . . . 10 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
10	2, 8, 9	syl2an 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)
11	5, 4, 7, 3, 10	bnj1177 32986	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑅 Fr 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅ ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ∈ V))
12		bnj1154 32979	. . . . . . . 8 ⊢ ((𝑅 Fr 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅ ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ∈ V) → ∃𝑢 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)∀𝑣 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ¬ 𝑣𝑅𝑢)
13	11, 12	bnj1176 32985	. . . . . . 7 ⊢ ∃𝑢∀𝑣((𝜑 ∧ 𝜓) → (𝑢 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ∧ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → (𝑣𝑅𝑢 → ¬ 𝑣 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)))))
14		biid 260	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑢)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑢)))
15		biid 260	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴))
16	4, 14, 15	bnj1175 32984	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → (𝑣𝑅𝑢 → 𝑣 ∈ trCl(𝑥, 𝐴, 𝑅)))
17	4, 13, 16	bnj1174 32983	. . . . . 6 ⊢ ∃𝑢∀𝑣((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑢 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)) ∧ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → (𝑣𝑅𝑢 → ¬ 𝑣 ∈ 𝐵))))
18	4, 15, 7, 10	bnj1173 32982	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑢 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)) → (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ↔ 𝑣 ∈ 𝐴))
19	4, 17, 18	bnj1172 32981	. . . . 5 ⊢ ∃𝑢∀𝑣((𝜑 ∧ 𝜓) → (𝑢 ∈ 𝐵 ∧ (𝑣 ∈ 𝐴 → (𝑣𝑅𝑢 → ¬ 𝑣 ∈ 𝐵))))
20	3, 19	bnj1171 32980	. . . 4 ⊢ ∃𝑢∀𝑣((𝜑 ∧ 𝜓) → (𝑢 ∈ 𝐵 ∧ (𝑣 ∈ 𝐵 → ¬ 𝑣𝑅𝑢)))
21	20	bnj1186 32987	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ¬ 𝑣𝑅𝑢)
22	21	bnj1185 32773	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
23	22	bnj1185 32773	1 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑤 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑤)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256   class class class wbr 5074   FrSe w-bnj15 32671   trClc-bnj18 32673

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-bnj17 32666  df-bnj14 32668  df-bnj13 32670  df-bnj15 32672  df-bnj18 32674  df-bnj19 32676

This theorem is referenced by:  bnj1189  32989