Theorem bnj1001 32839

Description: Technical lemma for bnj69 32890. 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 497	. . . 4 ⊢ (𝜂 → 𝑖 ∈ 𝑛)
4	3	bnj708 32636	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑖 ∈ 𝑛)
5		bnj1001.3	. . . . . 6 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	5	bnj1232 32683	. . . . 5 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
7	6	bnj706 32634	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑛 ∈ 𝐷)
8		bnj1001.13	. . . . 5 ⊢ 𝐷 = (ω ∖ {∅})
9	8	bnj923 32648	. . . 4 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
10	7, 9	syl 17	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑛 ∈ ω)
11		elnn 7698	. . 3 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑖 ∈ ω)
12	4, 10, 11	syl2anc 583	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑖 ∈ ω)
13		bnj1001.5	. . . . . 6 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
14	13	simp3bi 1145	. . . . 5 ⊢ (𝜏 → 𝑝 = suc 𝑛)
15	14	bnj707 32635	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑝 = suc 𝑛)
16		nnord 7695	. . . . . . 7 ⊢ (𝑛 ∈ ω → Ord 𝑛)
17		ordsucelsuc 7644	. . . . . . 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 476	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → suc 𝑖 ∈ suc 𝑛)
20		eleq2 2827	. . . . 5 ⊢ (𝑝 = suc 𝑛 → (suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
21	19, 20	anim12i 612	. . . 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 834	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
23		bianir 1055	. . 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 1126	1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880  ∅c0 4253  {csn 4558  Ord word 6250  suc csuc 6253   Fn wfn 6413  ‘cfv 6418  ωcom 7687   ∧ w-bnj17 32565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-om 7688  df-bnj17 32566

This theorem is referenced by:  bnj1020  32845