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Theorem bnj1279 31414
Description: Technical lemma for bnj60 31458. 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 4074 . . . . . . . 8 (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸))
2 elin 3939 . . . . . . . . 9 (𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ↔ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦𝐸))
32exbii 1923 . . . . . . . 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 31085 . . . . . . . . 9 pred(𝑥, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑥}
65bnj1538 31253 . . . . . . . 8 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
76anim1i 593 . . . . . . 7 ((𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦𝐸) → (𝑦𝑅𝑥𝑦𝐸))
84, 7bnj593 31143 . . . . . 6 (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ → ∃𝑦(𝑦𝑅𝑥𝑦𝐸))
983ad2ant3 1130 . . . . 5 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∃𝑦(𝑦𝑅𝑥𝑦𝐸))
10 nfv 1992 . . . . . . 7 𝑦 𝑥𝐸
11 nfra1 3079 . . . . . . 7 𝑦𝑦𝐸 ¬ 𝑦𝑅𝑥
12 nfv 1992 . . . . . . 7 𝑦( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅
1310, 11, 12nf3an 1980 . . . . . 6 𝑦(𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
1413nf5ri 2212 . . . . 5 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∀𝑦(𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅))
159, 14bnj1275 31212 . . . 4 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∃𝑦((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸))
16 simp2 1132 . . . 4 (((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸) → 𝑦𝑅𝑥)
17 simp12 1247 . . . . 5 (((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸) → ∀𝑦𝐸 ¬ 𝑦𝑅𝑥)
18 simp3 1133 . . . . 5 (((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸) → 𝑦𝐸)
1917, 18bnj1294 31216 . . . 4 (((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸) → ¬ 𝑦𝑅𝑥)
2015, 16, 19bnj1304 31218 . . 3 ¬ (𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
2120bnj1224 31200 . 2 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥) → ¬ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
22 nne 2936 . 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 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wne 2932  wral 3050  wrex 3051  {crab 3054  cin 3714  wss 3715  c0 4058  cop 4327   class class class wbr 4804  dom cdm 5266  cres 5268   Fn wfn 6044  cfv 6049  w-bnj17 31082   predc-bnj14 31084   FrSe w-bnj15 31088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-in 3722  df-nul 4059  df-bnj14 31085
This theorem is referenced by:  bnj1311  31420
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