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Theorem bnj1309 31216
Description: Technical lemma for bnj60 31256. 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 2971 . . . 4 (∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 → ∀𝑥𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
32bnj1352 31024 . . 3 ((𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) → ∀𝑥(𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
43hbab 2642 . 2 (𝑤 ∈ {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} → ∀𝑥 𝑤 ∈ {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)})
51, 4hbxfreq 2759 1 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wss 3607   predc-bnj14 30882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-ral 2946
This theorem is referenced by:  bnj1311  31218  bnj1373  31224  bnj1498  31255  bnj1525  31263  bnj1523  31265
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