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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 ↔ (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 496 . . . 4 (𝜂𝑖𝑛)
43bnj708 34601 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑖𝑛)
5 bnj1001.3 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
65bnj1232 34648 . . . . 5 (𝜒𝑛𝐷)
76bnj706 34599 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝑛𝐷)
8 bnj1001.13 . . . . 5 𝐷 = (ω ∖ {∅})
98bnj923 34613 . . . 4 (𝑛𝐷𝑛 ∈ ω)
107, 9syl 17 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑛 ∈ ω)
11 elnn 7887 . . 3 ((𝑖𝑛𝑛 ∈ ω) → 𝑖 ∈ ω)
124, 10, 11syl2anc 582 . 2 ((𝜃𝜒𝜏𝜂) → 𝑖 ∈ ω)
13 bnj1001.5 . . . . . 6 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
1413simp3bi 1144 . . . . 5 (𝜏𝑝 = suc 𝑛)
1514bnj707 34600 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝑝 = suc 𝑛)
16 nnord 7884 . . . . . . 7 (𝑛 ∈ ω → Ord 𝑛)
17 ordsucelsuc 7831 . . . . . . 7 (Ord 𝑛 → (𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
189, 16, 173syl 18 . . . . . 6 (𝑛𝐷 → (𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
1918biimpa 475 . . . . 5 ((𝑛𝐷𝑖𝑛) → suc 𝑖 ∈ suc 𝑛)
20 eleq2 2815 . . . . 5 (𝑝 = suc 𝑛 → (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
2119, 20anim12i 611 . . . 4 (((𝑛𝐷𝑖𝑛) ∧ 𝑝 = suc 𝑛) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
227, 4, 15, 21syl21anc 836 . . 3 ((𝜃𝜒𝜏𝜂) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
23 bianir 1056 . . 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 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
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