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Theorem bnj1309 32573
Description: Technical lemma for bnj60 32613. 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 3132 . . . 4 (∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 → ∀𝑥𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
32bnj1352 32378 . . 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 2861 1 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1540   = wceq 1542  wcel 2114  {cab 2716  wral 3053  wss 3843   predc-bnj14 32237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058
This theorem is referenced by:  bnj1311  32575  bnj1373  32581  bnj1498  32612  bnj1525  32620  bnj1523  32622
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