Theorem bnj556 35530

Description: Technical lemma for bnj852 35551. 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.)

Hypotheses

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

Assertion

Ref	Expression
bnj556	⊢ (𝜂 → 𝜎)

Proof of Theorem bnj556

Step	Hyp	Ref	Expression
1		vex 3454	. . . . 5 ⊢ 𝑝 ∈ V
2	1	bnj216 35363	. . . 4 ⊢ (𝑚 = suc 𝑝 → 𝑝 ∈ 𝑚)
3	2	3anim3i 1172	. . 3 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝) → (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
4	3	adantr 486	. 2 ⊢ (((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω) → (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
5		bnj556.19	. . 3 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
6		bnj258 35339	. . 3 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω))
7	5, 6	bitri 278	. 2 ⊢ (𝜂 ↔ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω))
8		bnj556.18	. 2 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
9	4, 7, 8	3imtr4i 295	1 ⊢ (𝜂 → 𝜎)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  suc csuc 6364  ωcom 7877   ∧ w-bnj17 35317

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 4151  df-sn 4584  df-suc 6368  df-bnj17 35318

This theorem is used by:  bnj557  35531  bnj561  35533  bnj562  35534