Theorem bnj556

Description: Technical lemma for bnj852 . 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.)

	Ref	Expression
Hypotheses	bnj556.18	⊢ ( 𝜎 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
	bnj556.19	⊢ ( 𝜂 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝 ) )
Assertion	bnj556	⊢ ( 𝜂 → 𝜎 )

Proof

Step	Hyp	Ref	Expression
1		bnj556.18	⊢ ( 𝜎 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
2		bnj556.19	⊢ ( 𝜂 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝 ) )
3		vex	⊢ 𝑝 ∈ V
4	3	bnj216	⊢ ( 𝑚 = suc 𝑝 → 𝑝 ∈ 𝑚 )
5	4	3anim3i	⊢ ( ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝 ) → ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
6	5	adantr	⊢ ( ( ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝 ) ∧ 𝑝 ∈ ω ) → ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
7		bnj258	⊢ ( ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝 ) ↔ ( ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝 ) ∧ 𝑝 ∈ ω ) )
8	2 7	bitri	⊢ ( 𝜂 ↔ ( ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝 ) ∧ 𝑝 ∈ ω ) )
9	6 8 1	3imtr4i	⊢ ( 𝜂 → 𝜎 )