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Theorem bnj1279 30794
Description: Technical lemma for bnj60 30838. 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 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1279.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1279.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1279.4 𝐷 = (dom 𝑔 ∩ dom )
bnj1279.5 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
bnj1279.6 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
bnj1279.7 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
Assertion
Ref Expression
bnj1279 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 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,,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐴(𝑥,𝑓,𝑔,,𝑑)   𝐵(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐷(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝑅(𝑥,𝑓,𝑔,,𝑑)   𝐸(𝑥,𝑓,𝑔,,𝑑)   𝐺(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑔,,𝑑)

Proof of Theorem bnj1279
StepHypRef Expression
1 n0 3907 . . . . . . . 8 (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸))
2 elin 3774 . . . . . . . . 9 (𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ↔ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦𝐸))
32exbii 1771 . . . . . . . 8 (∃𝑦 𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ↔ ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦𝐸))
41, 3sylbb 209 . . . . . . 7 (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦𝐸))
5 df-bnj14 30462 . . . . . . . . 9 pred(𝑥, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑥}
65bnj1538 30633 . . . . . . . 8 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
76anim1i 591 . . . . . . 7 ((𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦𝐸) → (𝑦𝑅𝑥𝑦𝐸))
84, 7bnj593 30523 . . . . . 6 (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ → ∃𝑦(𝑦𝑅𝑥𝑦𝐸))
983ad2ant3 1082 . . . . 5 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∃𝑦(𝑦𝑅𝑥𝑦𝐸))
10 nfv 1840 . . . . . . 7 𝑦 𝑥𝐸
11 nfra1 2936 . . . . . . 7 𝑦𝑦𝐸 ¬ 𝑦𝑅𝑥
12 nfv 1840 . . . . . . 7 𝑦( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅
1310, 11, 12nf3an 1828 . . . . . 6 𝑦(𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
1413nf5ri 2063 . . . . 5 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∀𝑦(𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅))
159, 14bnj1275 30592 . . . 4 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∃𝑦((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸))
16 simp2 1060 . . . 4 (((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸) → 𝑦𝑅𝑥)
17 simp12 1090 . . . . 5 (((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸) → ∀𝑦𝐸 ¬ 𝑦𝑅𝑥)
18 simp3 1061 . . . . 5 (((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸) → 𝑦𝐸)
1917, 18bnj1294 30596 . . . 4 (((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸) → ¬ 𝑦𝑅𝑥)
2015, 16, 19bnj1304 30598 . . 3 ¬ (𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
2120bnj1224 30580 . 2 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥) → ¬ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
22 nne 2794 . 2 (¬ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ ↔ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
2321, 22sylib 208 1 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  {crab 2911  cin 3554  wss 3555  c0 3891  cop 4154   class class class wbr 4613  dom cdm 5074  cres 5076   Fn wfn 5842  cfv 5847  w-bnj17 30459   predc-bnj14 30461   FrSe w-bnj15 30465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-in 3562  df-nul 3892  df-bnj14 30462
This theorem is referenced by:  bnj1311  30800
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