Theorem bnj93 35359

Description: Technical lemma for bnj97 35362. 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 35190	. . . 4 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
2	1	simprbi 503	. . 3 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Se 𝐴)
3		df-bnj13 35188	. . 3 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
4	2, 3	sylib 221	. 2 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
5	4	r19.21bi 3256	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   Fr wfr 5609   predc-bnj14 35185   Se w-bnj13 35187   FrSe w-bnj15 35189

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ral 3079  df-bnj13 35188  df-bnj15 35190

This theorem is used by:  bnj96  35361  bnj97  35362  bnj149  35371  bnj150  35372  bnj518  35382  bnj1148  35492