Theorem bnj1001 30282

Description: Technical lemma for bnj69 30332. 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 475	. . . 4 ⊢ (𝜂 → 𝑖 ∈ 𝑛)
4	3	bnj708 30080	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑖 ∈ 𝑛)
5		bnj1001.3	. . . . . 6 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	5	bnj1232 30128	. . . . 5 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
7	6	bnj706 30078	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑛 ∈ 𝐷)
8		bnj1001.13	. . . . 5 ⊢ 𝐷 = (ω ∖ {∅})
9	8	bnj923 30092	. . . 4 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
10	7, 9	syl 17	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑛 ∈ ω)
11		elnn 6967	. . 3 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑖 ∈ ω)
12	4, 10, 11	syl2anc 691	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑖 ∈ ω)
13		bnj1001.5	. . . . . 6 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
14	13	simp3bi 1071	. . . . 5 ⊢ (𝜏 → 𝑝 = suc 𝑛)
15	14	bnj707 30079	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑝 = suc 𝑛)
16		nnord 6965	. . . . . . 7 ⊢ (𝑛 ∈ ω → Ord 𝑛)
17		ordsucelsuc 6914	. . . . . . 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 500	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → suc 𝑖 ∈ suc 𝑛)
20		eleq2 2677	. . . . 5 ⊢ (𝑝 = suc 𝑛 → (suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
21	19, 20	anim12i 588	. . . 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 1317	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
23		bianir 1001	. . 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 1235	1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  ∅c0 3874  {csn 4125  Ord word 5639  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957   ∧ w-bnj17 30005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958  df-bnj17 30006

This theorem is referenced by:  bnj1020  30287