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Theorem bnj1001 34498
Description: Technical lemma for bnj69 34549. 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 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1001.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj1001.6 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
bnj1001.13 𝐷 = (ω ∖ {∅})
bnj1001.27 ((𝜃𝜒𝜏𝜂) → 𝜒″)
Assertion
Ref Expression
bnj1001 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))

Proof of Theorem bnj1001
StepHypRef Expression
1 bnj1001.27 . 2 ((𝜃𝜒𝜏𝜂) → 𝜒″)
2 bnj1001.6 . . . . 5 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
32simplbi 497 . . . 4 (𝜂𝑖𝑛)
43bnj708 34295 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑖𝑛)
5 bnj1001.3 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
65bnj1232 34342 . . . . 5 (𝜒𝑛𝐷)
76bnj706 34293 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝑛𝐷)
8 bnj1001.13 . . . . 5 𝐷 = (ω ∖ {∅})
98bnj923 34307 . . . 4 (𝑛𝐷𝑛 ∈ ω)
107, 9syl 17 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑛 ∈ ω)
11 elnn 7862 . . 3 ((𝑖𝑛𝑛 ∈ ω) → 𝑖 ∈ ω)
124, 10, 11syl2anc 583 . 2 ((𝜃𝜒𝜏𝜂) → 𝑖 ∈ ω)
13 bnj1001.5 . . . . . 6 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
1413simp3bi 1144 . . . . 5 (𝜏𝑝 = suc 𝑛)
1514bnj707 34294 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝑝 = suc 𝑛)
16 nnord 7859 . . . . . . 7 (𝑛 ∈ ω → Ord 𝑛)
17 ordsucelsuc 7806 . . . . . . 7 (Ord 𝑛 → (𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
189, 16, 173syl 18 . . . . . 6 (𝑛𝐷 → (𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
1918biimpa 476 . . . . 5 ((𝑛𝐷𝑖𝑛) → suc 𝑖 ∈ suc 𝑛)
20 eleq2 2816 . . . . 5 (𝑝 = suc 𝑛 → (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
2119, 20anim12i 612 . . . 4 (((𝑛𝐷𝑖𝑛) ∧ 𝑝 = suc 𝑛) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
227, 4, 15, 21syl21anc 835 . . 3 ((𝜃𝜒𝜏𝜂) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
23 bianir 1055 . . 3 ((suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)) → suc 𝑖𝑝)
2422, 23syl 17 . 2 ((𝜃𝜒𝜏𝜂) → suc 𝑖𝑝)
251, 12, 243jca 1125 1 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  cdif 3940  c0 4317  {csn 4623  Ord word 6356  suc csuc 6359   Fn wfn 6531  cfv 6536  ωcom 7851  w-bnj17 34225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-om 7852  df-bnj17 34226
This theorem is referenced by:  bnj1020  34504
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