Theorem bnj556 32176

Description: Technical lemma for bnj852 32197. 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 3500	. . . . 5 ⊢ 𝑝 ∈ V
2	1	bnj216 32006	. . . 4 ⊢ (𝑚 = suc 𝑝 → 𝑝 ∈ 𝑚)
3	2	3anim3i 1150	. . 3 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝) → (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
4	3	adantr 483	. 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 31982	. . 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 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  suc csuc 6196  ωcom 7583   ∧ w-bnj17 31960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-un 3944  df-sn 4571  df-suc 6200  df-bnj17 31961

This theorem is referenced by:  bnj557  32177  bnj561  32179  bnj562  32180