Theorem bnj1190 35305

Description: Technical lemma for bnj69 35307. 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 1160	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
3	2	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴)
4		eqid 2764	. . . . . 6 ⊢ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) = ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)
5		bnj1190.2	. . . . . . . . 9 ⊢ (𝜓 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑥))
6	1	simp1bi 1159	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 FrSe 𝐴)
7	6	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 FrSe 𝐴)
8	5	simp1bi 1159	. . . . . . . . . 10 ⊢ (𝜓 → 𝑥 ∈ 𝐵)
9		ssel2 3933	. . . . . . . . . 10 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
10	2, 8, 9	syl2an 605	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)
11	5, 4, 7, 3, 10	bnj1177 35303	. . . . . . . 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 35296	. . . . . . . 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 35302	. . . . . . 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 263	. . . . . . . 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 263	. . . . . . . 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 35301	. . . . . . 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 35300	. . . . . 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 35299	. . . . . 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 35298	. . . . 5 ⊢ ∃𝑢∀𝑣((𝜑 ∧ 𝜓) → (𝑢 ∈ 𝐵 ∧ (𝑣 ∈ 𝐴 → (𝑣𝑅𝑢 → ¬ 𝑣 ∈ 𝐵))))
20	3, 19	bnj1171 35297	. . . 4 ⊢ ∃𝑢∀𝑣((𝜑 ∧ 𝜓) → (𝑢 ∈ 𝐵 ∧ (𝑣 ∈ 𝐵 → ¬ 𝑣𝑅𝑢)))
21	20	bnj1186 35304	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ¬ 𝑣𝑅𝑢)
22	21	bnj1185 35090	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
23	22	bnj1185 35090	1 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑤 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑤)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078  ∃wrex 3088   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287   class class class wbr 5102   FrSe w-bnj15 34990   trClc-bnj18 34992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-reg 9542  ax-inf2 9598

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-om 7849  df-1o 8439  df-bnj17 34985  df-bnj14 34987  df-bnj13 34989  df-bnj15 34991  df-bnj18 34993  df-bnj19 34995

This theorem is referenced by:  bnj1189  35306