Theorem bnj1309 32902

Description: Technical lemma for bnj60 32942. 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 3143	. . . 4 ⊢ (∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 → ∀𝑥∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
3	2	bnj1352 32707	. . 3 ⊢ ((𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) → ∀𝑥(𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
4	3	hbab 2726	. 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 2868	1 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063   ⊆ wss 3883   predc-bnj14 32567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068

This theorem is referenced by:  bnj1311  32904  bnj1373  32910  bnj1498  32941  bnj1525  32949  bnj1523  32951