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Theorem bnj970 31486
Description: Technical lemma for bnj69 31547. 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 31343 . . . 4 (𝜒𝑛𝐷)
323ad2ant1 1163 . . 3 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑛𝐷)
43adantl 473 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑛𝐷)
5 simpr3 1252 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛)
6 bnj970.10 . . . . 5 𝐷 = (ω ∖ {∅})
76bnj923 31307 . . . 4 (𝑛𝐷𝑛 ∈ ω)
8 peano2 7288 . . . . 5 (𝑛 ∈ ω → suc 𝑛 ∈ ω)
9 eleq1 2832 . . . . 5 (𝑝 = suc 𝑛 → (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω))
10 bianir 1081 . . . . 5 ((suc 𝑛 ∈ ω ∧ (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω)) → 𝑝 ∈ ω)
118, 9, 10syl2an 589 . . . 4 ((𝑛 ∈ ω ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω)
127, 11sylan 575 . . 3 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝 ∈ ω)
13 df-suc 5916 . . . . . 6 suc 𝑛 = (𝑛 ∪ {𝑛})
1413eqeq2i 2777 . . . . 5 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
15 ssun2 3941 . . . . . . 7 {𝑛} ⊆ (𝑛 ∪ {𝑛})
16 vex 3353 . . . . . . . 8 𝑛 ∈ V
1716snnz 4465 . . . . . . 7 {𝑛} ≠ ∅
18 ssn0 4140 . . . . . . 7 (({𝑛} ⊆ (𝑛 ∪ {𝑛}) ∧ {𝑛} ≠ ∅) → (𝑛 ∪ {𝑛}) ≠ ∅)
1915, 17, 18mp2an 683 . . . . . 6 (𝑛 ∪ {𝑛}) ≠ ∅
20 neeq1 2999 . . . . . 6 (𝑝 = (𝑛 ∪ {𝑛}) → (𝑝 ≠ ∅ ↔ (𝑛 ∪ {𝑛}) ≠ ∅))
2119, 20mpbiri 249 . . . . 5 (𝑝 = (𝑛 ∪ {𝑛}) → 𝑝 ≠ ∅)
2214, 21sylbi 208 . . . 4 (𝑝 = suc 𝑛𝑝 ≠ ∅)
2322adantl 473 . . 3 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝 ≠ ∅)
246eleq2i 2836 . . . 4 (𝑝𝐷𝑝 ∈ (ω ∖ {∅}))
25 eldifsn 4474 . . . 4 (𝑝 ∈ (ω ∖ {∅}) ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
2624, 25bitri 266 . . 3 (𝑝𝐷 ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
2712, 23, 26sylanbrc 578 . 2 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝𝐷)
284, 5, 27syl2anc 579 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  cdif 3731  cun 3732  wss 3734  c0 4081  {csn 4336  suc csuc 5912   Fn wfn 6065  ωcom 7267  w-bnj17 31224   FrSe w-bnj15 31230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-tr 4914  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-om 7268  df-bnj17 31225
This theorem is referenced by:  bnj910  31487
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