Theorem bnj1279 35514

Description: Technical lemma for bnj60 35558. 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 4303	. . . . . . . 8 ⊢ (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸))
2		elin 3918	. . . . . . . . 9 ⊢ (𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ↔ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦 ∈ 𝐸))
3	2	exbii 1881	. . . . . . . 8 ⊢ (∃𝑦 𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ↔ ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦 ∈ 𝐸))
4	1, 3	sylbb 222	. . . . . . 7 ⊢ (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦 ∈ 𝐸))
5		df-bnj14 35186	. . . . . . . . 9 ⊢ pred(𝑥, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥}
6	5	bnj1538 35351	. . . . . . . 8 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
7	6	anim1i 627	. . . . . . 7 ⊢ ((𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦 ∈ 𝐸) → (𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸))
8	4, 7	bnj593 35242	. . . . . 6 ⊢ (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ → ∃𝑦(𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸))
9	8	3ad2ant3 1153	. . . . 5 ⊢ ((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∃𝑦(𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸))
10		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐸
11		nfra1 3288	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥
12		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑦( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅
13	10, 11, 12	nf3an 1934	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
14	13	nf5ri 2233	. . . . 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 35309	. . . 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 1155	. . . 4 ⊢ (((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸) → 𝑦𝑅𝑥)
17		simp12 1223	. . . . 5 ⊢ (((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸) → ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥)
18		simp3 1156	. . . . 5 ⊢ (((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸) → 𝑦 ∈ 𝐸)
19	17, 18	bnj1294 35313	. . . 4 ⊢ (((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥 ∧ 𝑦 ∈ 𝐸) → ¬ 𝑦𝑅𝑥)
20	15, 16, 19	bnj1304 35315	. . 3 ⊢ ¬ (𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
21	20	bnj1224 35297	. 2 ⊢ ((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥) → ¬ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
22		nne 2961	. 2 ⊢ (¬ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ ↔ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
23	21, 22	sylib 221	1 ⊢ ((𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2740   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ⟨cop 4593   class class class wbr 5107  dom cdm 5659   ↾ cres 5661   Fn wfn 6532  ‘cfv 6537   ∧ w-bnj17 35183   predc-bnj14 35185   FrSe w-bnj15 35189

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-in 3909  df-nul 4283  df-bnj14 35186

This theorem is used by:  bnj1311  35520