Theorem bnj1190 35022

Description: Technical lemma for bnj69 35024. 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 1147	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
3	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴)
4		eqid 2737	. . . . . 6 ⊢ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) = ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)
5		bnj1190.2	. . . . . . . . 9 ⊢ (𝜓 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑥))
6	1	simp1bi 1146	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 FrSe 𝐴)
7	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 FrSe 𝐴)
8	5	simp1bi 1146	. . . . . . . . . 10 ⊢ (𝜓 → 𝑥 ∈ 𝐵)
9		ssel2 3978	. . . . . . . . . 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 35020	. . . . . . . 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 35013	. . . . . . . 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 35019	. . . . . . 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 261	. . . . . . . 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 261	. . . . . . . 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 35018	. . . . . . 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 35017	. . . . . 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 35016	. . . . . 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 35015	. . . . 5 ⊢ ∃𝑢∀𝑣((𝜑 ∧ 𝜓) → (𝑢 ∈ 𝐵 ∧ (𝑣 ∈ 𝐴 → (𝑣𝑅𝑢 → ¬ 𝑣 ∈ 𝐵))))
20	3, 19	bnj1171 35014	. . . 4 ⊢ ∃𝑢∀𝑣((𝜑 ∧ 𝜓) → (𝑢 ∈ 𝐵 ∧ (𝑣 ∈ 𝐵 → ¬ 𝑣𝑅𝑢)))
21	20	bnj1186 35021	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ¬ 𝑣𝑅𝑢)
22	21	bnj1185 34807	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
23	22	bnj1185 34807	1 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑤 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑤)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143   FrSe w-bnj15 34706   trClc-bnj18 34708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-bnj17 34701  df-bnj14 34703  df-bnj13 34705  df-bnj15 34707  df-bnj18 34709  df-bnj19 34711

This theorem is referenced by:  bnj1189  35023