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Theorem bnj1279 32298
Description: Technical lemma for bnj60 32342. 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 4286 . . . . . . . 8 (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸))
2 elin 3929 . . . . . . . . 9 (𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ↔ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦𝐸))
32exbii 1848 . . . . . . . 8 (∃𝑦 𝑦 ∈ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ↔ ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦𝐸))
41, 3sylbb 221 . . . . . . 7 (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦𝐸))
5 df-bnj14 31967 . . . . . . . . 9 pred(𝑥, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑥}
65bnj1538 32135 . . . . . . . 8 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
76anim1i 616 . . . . . . 7 ((𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑦𝐸) → (𝑦𝑅𝑥𝑦𝐸))
84, 7bnj593 32024 . . . . . 6 (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ → ∃𝑦(𝑦𝑅𝑥𝑦𝐸))
983ad2ant3 1131 . . . . 5 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∃𝑦(𝑦𝑅𝑥𝑦𝐸))
10 nfv 1915 . . . . . . 7 𝑦 𝑥𝐸
11 nfra1 3206 . . . . . . 7 𝑦𝑦𝐸 ¬ 𝑦𝑅𝑥
12 nfv 1915 . . . . . . 7 𝑦( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅
1310, 11, 12nf3an 1902 . . . . . 6 𝑦(𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
1413nf5ri 2195 . . . . 5 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∀𝑦(𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅))
159, 14bnj1275 32093 . . . 4 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) → ∃𝑦((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸))
16 simp2 1133 . . . 4 (((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸) → 𝑦𝑅𝑥)
17 simp12 1200 . . . . 5 (((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸) → ∀𝑦𝐸 ¬ 𝑦𝑅𝑥)
18 simp3 1134 . . . . 5 (((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸) → 𝑦𝐸)
1917, 18bnj1294 32097 . . . 4 (((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅) ∧ 𝑦𝑅𝑥𝑦𝐸) → ¬ 𝑦𝑅𝑥)
2015, 16, 19bnj1304 32099 . . 3 ¬ (𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥 ∧ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
2120bnj1224 32081 . 2 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥) → ¬ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅)
22 nne 3010 . 2 (¬ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) ≠ ∅ ↔ ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
2321, 22sylib 220 1 ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wex 1780  wcel 2114  {cab 2798  wne 3006  wral 3125  wrex 3126  {crab 3129  cin 3912  wss 3913  c0 4269  cop 4549   class class class wbr 5042  dom cdm 5531  cres 5533   Fn wfn 6326  cfv 6331  w-bnj17 31964   predc-bnj14 31966   FrSe w-bnj15 31970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rab 3134  df-v 3475  df-dif 3916  df-in 3920  df-nul 4270  df-bnj14 31967
This theorem is referenced by:  bnj1311  32304
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