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Theorem bnj970 32119
Description: Technical lemma for bnj69 32180. 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
bnj970.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj970.10 𝐷 = (ω ∖ {∅})
Assertion
Ref Expression
bnj970 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝𝐷)

Proof of Theorem bnj970
StepHypRef Expression
1 bnj970.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
21bnj1232 31975 . . . 4 (𝜒𝑛𝐷)
323ad2ant1 1125 . . 3 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑛𝐷)
43adantl 482 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑛𝐷)
5 simpr3 1188 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛)
6 bnj970.10 . . . . 5 𝐷 = (ω ∖ {∅})
76bnj923 31939 . . . 4 (𝑛𝐷𝑛 ∈ ω)
8 peano2 7590 . . . . 5 (𝑛 ∈ ω → suc 𝑛 ∈ ω)
9 eleq1 2900 . . . . 5 (𝑝 = suc 𝑛 → (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω))
10 bianir 1050 . . . . 5 ((suc 𝑛 ∈ ω ∧ (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω)) → 𝑝 ∈ ω)
118, 9, 10syl2an 595 . . . 4 ((𝑛 ∈ ω ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω)
127, 11sylan 580 . . 3 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝 ∈ ω)
13 df-suc 6191 . . . . . 6 suc 𝑛 = (𝑛 ∪ {𝑛})
1413eqeq2i 2834 . . . . 5 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
15 ssun2 4148 . . . . . . 7 {𝑛} ⊆ (𝑛 ∪ {𝑛})
16 vex 3498 . . . . . . . 8 𝑛 ∈ V
1716snnz 4705 . . . . . . 7 {𝑛} ≠ ∅
18 ssn0 4353 . . . . . . 7 (({𝑛} ⊆ (𝑛 ∪ {𝑛}) ∧ {𝑛} ≠ ∅) → (𝑛 ∪ {𝑛}) ≠ ∅)
1915, 17, 18mp2an 688 . . . . . 6 (𝑛 ∪ {𝑛}) ≠ ∅
20 neeq1 3078 . . . . . 6 (𝑝 = (𝑛 ∪ {𝑛}) → (𝑝 ≠ ∅ ↔ (𝑛 ∪ {𝑛}) ≠ ∅))
2119, 20mpbiri 259 . . . . 5 (𝑝 = (𝑛 ∪ {𝑛}) → 𝑝 ≠ ∅)
2214, 21sylbi 218 . . . 4 (𝑝 = suc 𝑛𝑝 ≠ ∅)
2322adantl 482 . . 3 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝 ≠ ∅)
246eleq2i 2904 . . . 4 (𝑝𝐷𝑝 ∈ (ω ∖ {∅}))
25 eldifsn 4713 . . . 4 (𝑝 ∈ (ω ∖ {∅}) ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
2624, 25bitri 276 . . 3 (𝑝𝐷 ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
2712, 23, 26sylanbrc 583 . 2 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝𝐷)
284, 5, 27syl2anc 584 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3016  cdif 3932  cun 3933  wss 3935  c0 4290  {csn 4559  suc csuc 6187   Fn wfn 6344  ωcom 7568  w-bnj17 31856   FrSe w-bnj15 31862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-tr 5165  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-om 7569  df-bnj17 31857
This theorem is referenced by:  bnj910  32120
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