Theorem bnj1001 35509

Description: Technical lemma for bnj69 35560. 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
bnj1001.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1001.5	⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
bnj1001.6	⊢ (𝜂 ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
bnj1001.13	⊢ 𝐷 = (ω ∖ {∅})
bnj1001.27	⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″)

Assertion

Ref	Expression
bnj1001	⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝))

Proof of Theorem bnj1001

Step	Hyp	Ref	Expression
1		bnj1001.27	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″)
2		bnj1001.6	. . . . 5 ⊢ (𝜂 ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
3	2	simplbi 502	. . . 4 ⊢ (𝜂 → 𝑖 ∈ 𝑛)
4	3	bnj708 35307	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑖 ∈ 𝑛)
5		bnj1001.3	. . . . . 6 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	5	bnj1232 35353	. . . . 5 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
7	6	bnj706 35305	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑛 ∈ 𝐷)
8		bnj1001.13	. . . . 5 ⊢ 𝐷 = (ω ∖ {∅})
9	8	bnj923 35319	. . . 4 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
10	7, 9	syl 18	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑛 ∈ ω)
11		elnn 7872	. . 3 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑖 ∈ ω)
12	4, 10, 11	syl2anc 596	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑖 ∈ ω)
13		bnj1001.5	. . . . . 6 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
14	13	simp3bi 1165	. . . . 5 ⊢ (𝜏 → 𝑝 = suc 𝑛)
15	14	bnj707 35306	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑝 = suc 𝑛)
16		nnord 7869	. . . . . . 7 ⊢ (𝑛 ∈ ω → Ord 𝑛)
17		ordsucelsuc 7817	. . . . . . 7 ⊢ (Ord 𝑛 → (𝑖 ∈ 𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
18	9, 16, 17	3syl 19	. . . . . 6 ⊢ (𝑛 ∈ 𝐷 → (𝑖 ∈ 𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
19	18	biimpa 482	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → suc 𝑖 ∈ suc 𝑛)
20		eleq2 2849	. . . . 5 ⊢ (𝑝 = suc 𝑛 → (suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
21	19, 20	anim12i 625	. . . 4 ⊢ (((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) ∧ 𝑝 = suc 𝑛) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
22	7, 4, 15, 21	syl21anc 851	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
23		bianir 1074	. . 3 ⊢ ((suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛)) → suc 𝑖 ∈ 𝑝)
24	22, 23	syl 18	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ 𝑝)
25	1, 12, 24	3jca 1146	1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896  ∅c0 4279  {csn 4584  Ord word 6351  suc csuc 6354   Fn wfn 6523  ‘cfv 6528  ωcom 7861   ∧ w-bnj17 35237

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5249  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-om 7862  df-bnj17 35238

This theorem is used by:  bnj1020  35515