Theorem bnj93 34856

Description: Technical lemma for bnj97 34859. 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.)

Assertion

Ref	Expression
bnj93	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem bnj93

Step	Hyp	Ref	Expression
1		df-bnj15 34686	. . . 4 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
2	1	simprbi 496	. . 3 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Se 𝐴)
3		df-bnj13 34684	. . 3 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
4	2, 3	sylib 218	. 2 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
5	4	r19.21bi 3249	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   Fr wfr 5638   predc-bnj14 34681   Se w-bnj13 34683   FrSe w-bnj15 34685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-ral 3060  df-bnj13 34684  df-bnj15 34686

This theorem is referenced by:  bnj96  34858  bnj97  34859  bnj149  34868  bnj150  34869  bnj518  34879  bnj1148  34989