Theorem bnj1279 32294

Description: Technical lemma for bnj60 32338. 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
bnj1279.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1279.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1279.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1279.4	⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
bnj1279.5	⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
bnj1279.6	⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
bnj1279.7	⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))

Assertion

Ref	Expression
bnj1279	⊢ ((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐸   𝑦,𝑅   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐴(𝑥,𝑓,𝑔,ℎ,𝑑)   𝐵(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐷(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝑅(𝑥,𝑓,𝑔,ℎ,𝑑)   𝐸(𝑥,𝑓,𝑔,ℎ,𝑑)   𝐺(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)

Proof of Theorem bnj1279

Step	Hyp	Ref	Expression
1		n0 4313	. . . . . . . 8 ⊢ (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸))
2		elin 4172	. . . . . . . . 9 ⊢ (𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ↔ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦 ∈ 𝐸))
3	2	exbii 1847	. . . . . . . 8 ⊢ (∃𝑦 𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ↔ ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦 ∈ 𝐸))
4	1, 3	sylbb 221	. . . . . . 7 ⊢ (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦 ∈ 𝐸))
5		df-bnj14 31963	. . . . . . . . 9 ⊢ pred(𝑥, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥}
6	5	bnj1538 32131	. . . . . . . 8 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
7	6	anim1i 616	. . . . . . 7 ⊢ ((𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦 ∈ 𝐸) → (𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸))
8	4, 7	bnj593 32020	. . . . . 6 ⊢ (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ → ∃𝑦(𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸))
9	8	3ad2ant3 1131	. . . . 5 ⊢ ((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∃𝑦(𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸))
10		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐸
11		nfra1 3222	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥
12		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑦( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅
13	10, 11, 12	nf3an 1901	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
14	13	nf5ri 2194	. . . . 5 ⊢ ((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∀𝑦(𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅))
15	9, 14	bnj1275 32089	. . . 4 ⊢ ((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∃𝑦((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸))
16		simp2 1133	. . . 4 ⊢ (((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸) → 𝑦𝑅𝑥)
17		simp12 1200	. . . . 5 ⊢ (((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸) → ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥)
18		simp3 1134	. . . . 5 ⊢ (((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸) → 𝑦 ∈ 𝐸)
19	17, 18	bnj1294 32093	. . . 4 ⊢ (((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸) → ¬ 𝑦𝑅𝑥)
20	15, 16, 19	bnj1304 32095	. . 3 ⊢ ¬ (𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
21	20	bnj1224 32077	. 2 ⊢ ((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥) → ¬ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
22		nne 3023	. 2 ⊢ (¬ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ ↔ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
23	21, 22	sylib 220	1 ⊢ ((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536  ∃wex 1779   ∈ wcel 2113  {cab 2802   ≠ wne 3019  ∀wral 3141  ∃wrex 3142  {crab 3145   ∩ cin 3938   ⊆ wss 3939  ∅c0 4294  ⟨cop 4576   class class class wbr 5069  dom cdm 5558   ↾ cres 5560   Fn wfn 6353  ‘cfv 6358   ∧ w-bnj17 31960   predc-bnj14 31962   FrSe w-bnj15 31966

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rab 3150  df-v 3499  df-dif 3942  df-in 3946  df-nul 4295  df-bnj14 31963

This theorem is referenced by:  bnj1311  32300