Theorem bnj1309 31090

Description: Technical lemma for bnj60 31130. 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 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}

Assertion

Ref	Expression
bnj1309	⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)

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

Step	Hyp	Ref	Expression
1		bnj1309.1	. 2 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2		hbra1 2942	. . . 4 ⊢ (∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 → ∀𝑥∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
3	2	bnj1352 30898	. . 3 ⊢ ((𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) → ∀𝑥(𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
4	3	hbab 2613	. 2 ⊢ (𝑤 ∈ {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} → ∀𝑥 𝑤 ∈ {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)})
5	1, 4	hbxfreq 2730	1 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912   ⊆ wss 3574   predc-bnj14 30754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917

This theorem is referenced by:  bnj1311  31092  bnj1373  31098  bnj1498  31129  bnj1525  31137  bnj1523  31139