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	$⊢ σ \leftrightarrow (m \in D \land n = suc m \land p \in m)$
	bnj556.19	$⊢ η \leftrightarrow (m \in D \land n = suc m \land p \in ω \land m = suc p)$
Assertion	bnj556	$⊢ η \to σ$

Proof

Step	Hyp	Ref	Expression
1		bnj556.18	$⊢ σ \leftrightarrow (m \in D \land n = suc m \land p \in m)$
2		bnj556.19	$⊢ η \leftrightarrow (m \in D \land n = suc m \land p \in ω \land m = suc p)$
3		vex	$⊢ p \in V$
4	3	bnj216	$⊢ m = suc p \to p \in m$
5	4	3anim3i	$⊢ (m \in D \land n = suc m \land m = suc p) \to (m \in D \land n = suc m \land p \in m)$
6	5	adantr	$⊢ ((m \in D \land n = suc m \land m = suc p) \land p \in ω) \to (m \in D \land n = suc m \land p \in m)$
7		bnj258	$⊢ (m \in D \land n = suc m \land p \in ω \land m = suc p) \leftrightarrow ((m \in D \land n = suc m \land m = suc p) \land p \in ω)$
8	2 7	bitri	$⊢ η \leftrightarrow ((m \in D \land n = suc m \land m = suc p) \land p \in ω)$
9	6 8 1	3imtr4i	$⊢ η \to σ$

Metamath Proof Explorer

Theorem bnj556

Proof