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Theorem bnj1309 33002
Description: Technical lemma for bnj60 33042. 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.)
Hypothesis
Ref Expression
bnj1309.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
Assertion
Ref Expression
bnj1309 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑑   𝑥,𝑤
Allowed substitution hints:   𝐴(𝑤,𝑑)   𝐵(𝑥,𝑤,𝑑)   𝑅(𝑥,𝑤,𝑑)

Proof of Theorem bnj1309
StepHypRef Expression
1 bnj1309.1 . 2 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2 hbra1 3145 . . . 4 (∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 → ∀𝑥𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
32bnj1352 32807 . . 3 ((𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) → ∀𝑥(𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
43hbab 2726 . 2 (𝑤 ∈ {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} → ∀𝑥 𝑤 ∈ {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)})
51, 4hbxfreq 2869 1 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1537   = wceq 1539  wcel 2106  {cab 2715  wral 3064  wss 3887   predc-bnj14 32667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069
This theorem is referenced by:  bnj1311  33004  bnj1373  33010  bnj1498  33041  bnj1525  33049  bnj1523  33051
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