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Theorem bnj1001 33628
Description: Technical lemma for bnj69 33679. 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 499 . . . 4 (𝜂𝑖𝑛)
43bnj708 33425 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑖𝑛)
5 bnj1001.3 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
65bnj1232 33472 . . . . 5 (𝜒𝑛𝐷)
76bnj706 33423 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝑛𝐷)
8 bnj1001.13 . . . . 5 𝐷 = (ω ∖ {∅})
98bnj923 33437 . . . 4 (𝑛𝐷𝑛 ∈ ω)
107, 9syl 17 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑛 ∈ ω)
11 elnn 7814 . . 3 ((𝑖𝑛𝑛 ∈ ω) → 𝑖 ∈ ω)
124, 10, 11syl2anc 585 . 2 ((𝜃𝜒𝜏𝜂) → 𝑖 ∈ ω)
13 bnj1001.5 . . . . . 6 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
1413simp3bi 1148 . . . . 5 (𝜏𝑝 = suc 𝑛)
1514bnj707 33424 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝑝 = suc 𝑛)
16 nnord 7811 . . . . . . 7 (𝑛 ∈ ω → Ord 𝑛)
17 ordsucelsuc 7758 . . . . . . 7 (Ord 𝑛 → (𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
189, 16, 173syl 18 . . . . . 6 (𝑛𝐷 → (𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
1918biimpa 478 . . . . 5 ((𝑛𝐷𝑖𝑛) → suc 𝑖 ∈ suc 𝑛)
20 eleq2 2823 . . . . 5 (𝑝 = suc 𝑛 → (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
2119, 20anim12i 614 . . . 4 (((𝑛𝐷𝑖𝑛) ∧ 𝑝 = suc 𝑛) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
227, 4, 15, 21syl21anc 837 . . 3 ((𝜃𝜒𝜏𝜂) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
23 bianir 1058 . . 3 ((suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)) → suc 𝑖𝑝)
2422, 23syl 17 . 2 ((𝜃𝜒𝜏𝜂) → suc 𝑖𝑝)
251, 12, 243jca 1129 1 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  cdif 3908  c0 4283  {csn 4587  Ord word 6317  suc csuc 6320   Fn wfn 6492  cfv 6497  ωcom 7803  w-bnj17 33355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-om 7804  df-bnj17 33356
This theorem is referenced by:  bnj1020  33634
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