Theorem bnj556 35197

Description: Technical lemma for bnj852 35218. 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 3460	. . . . 5 ⊢ 𝑝 ∈ V
2	1	bnj216 35030	. . . 4 ⊢ (𝑚 = suc 𝑝 → 𝑝 ∈ 𝑚)
3	2	3anim3i 1168	. . 3 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝) → (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
4	3	adantr 484	. 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 35006	. . 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 277	. 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 294	1 ⊢ (𝜂 → 𝜎)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  suc csuc 6350  ωcom 7848   ∧ w-bnj17 34984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-un 3911  df-sn 4585  df-suc 6354  df-bnj17 34985

This theorem is referenced by:  bnj557  35198  bnj561  35200  bnj562  35201