Theorem bnj1001 34804

Description: Technical lemma for bnj69 34855. 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 496	. . . 4 ⊢ (𝜂 → 𝑖 ∈ 𝑛)
4	3	bnj708 34601	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑖 ∈ 𝑛)
5		bnj1001.3	. . . . . 6 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	5	bnj1232 34648	. . . . 5 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
7	6	bnj706 34599	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑛 ∈ 𝐷)
8		bnj1001.13	. . . . 5 ⊢ 𝐷 = (ω ∖ {∅})
9	8	bnj923 34613	. . . 4 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
10	7, 9	syl 17	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑛 ∈ ω)
11		elnn 7887	. . 3 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑖 ∈ ω)
12	4, 10, 11	syl2anc 582	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑖 ∈ ω)
13		bnj1001.5	. . . . . 6 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
14	13	simp3bi 1144	. . . . 5 ⊢ (𝜏 → 𝑝 = suc 𝑛)
15	14	bnj707 34600	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑝 = suc 𝑛)
16		nnord 7884	. . . . . . 7 ⊢ (𝑛 ∈ ω → Ord 𝑛)
17		ordsucelsuc 7831	. . . . . . 7 ⊢ (Ord 𝑛 → (𝑖 ∈ 𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
18	9, 16, 17	3syl 18	. . . . . 6 ⊢ (𝑛 ∈ 𝐷 → (𝑖 ∈ 𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
19	18	biimpa 475	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → suc 𝑖 ∈ suc 𝑛)
20		eleq2 2815	. . . . 5 ⊢ (𝑝 = suc 𝑛 → (suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
21	19, 20	anim12i 611	. . . 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 836	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
23		bianir 1056	. . 3 ⊢ ((suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛)) → suc 𝑖 ∈ 𝑝)
24	22, 23	syl 17	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ 𝑝)
25	1, 12, 24	3jca 1125	1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ∖ cdif 3944  ∅c0 4325  {csn 4633  Ord word 6375  suc csuc 6378   Fn wfn 6549  ‘cfv 6554  ωcom 7876   ∧ w-bnj17 34531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-tr 5271  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-om 7877  df-bnj17 34532

This theorem is referenced by:  bnj1020  34810