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Theorem bnj1309 34561
Description: Technical lemma for bnj60 34601. 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 3292 . . . 4 (∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 → ∀𝑥𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
32bnj1352 34366 . . 3 ((𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) → ∀𝑥(𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
43hbab 2714 . 2 (𝑤 ∈ {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} → ∀𝑥 𝑤 ∈ {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)})
51, 4hbxfreq 2858 1 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1531   = wceq 1533  wcel 2098  {cab 2703  wral 3055  wss 3943   predc-bnj14 34227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056
This theorem is referenced by:  bnj1311  34563  bnj1373  34569  bnj1498  34600  bnj1525  34608  bnj1523  34610
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