Theorem bnj1190 34984

Description: Technical lemma for bnj69 34986. 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 1146	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
3	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴)
4		eqid 2740	. . . . . 6 ⊢ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) = ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)
5		bnj1190.2	. . . . . . . . 9 ⊢ (𝜓 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑥))
6	1	simp1bi 1145	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 FrSe 𝐴)
7	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 FrSe 𝐴)
8	5	simp1bi 1145	. . . . . . . . . 10 ⊢ (𝜓 → 𝑥 ∈ 𝐵)
9		ssel2 4003	. . . . . . . . . 10 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
10	2, 8, 9	syl2an 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)
11	5, 4, 7, 3, 10	bnj1177 34982	. . . . . . . 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 34975	. . . . . . . 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 34981	. . . . . . 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 34980	. . . . . . 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 34979	. . . . . 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 34978	. . . . . 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 34977	. . . . 5 ⊢ ∃𝑢∀𝑣((𝜑 ∧ 𝜓) → (𝑢 ∈ 𝐵 ∧ (𝑣 ∈ 𝐴 → (𝑣𝑅𝑢 → ¬ 𝑣 ∈ 𝐵))))
20	3, 19	bnj1171 34976	. . . 4 ⊢ ∃𝑢∀𝑣((𝜑 ∧ 𝜓) → (𝑢 ∈ 𝐵 ∧ (𝑣 ∈ 𝐵 → ¬ 𝑣𝑅𝑢)))
21	20	bnj1186 34983	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ¬ 𝑣𝑅𝑢)
22	21	bnj1185 34769	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
23	22	bnj1185 34769	1 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑤 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑤)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166   FrSe w-bnj15 34668   trClc-bnj18 34670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-reg 9661  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-bnj17 34663  df-bnj14 34665  df-bnj13 34667  df-bnj15 34669  df-bnj18 34671  df-bnj19 34673

This theorem is referenced by:  bnj1189  34985