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Theorem bnj970 34578
Description: Technical lemma for bnj69 34641. 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 34434 . . . 4 (𝜒𝑛𝐷)
323ad2ant1 1131 . . 3 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑛𝐷)
43adantl 481 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑛𝐷)
5 simpr3 1194 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛)
6 bnj970.10 . . . . 5 𝐷 = (ω ∖ {∅})
76bnj923 34399 . . . 4 (𝑛𝐷𝑛 ∈ ω)
8 peano2 7896 . . . . 5 (𝑛 ∈ ω → suc 𝑛 ∈ ω)
9 eleq1 2817 . . . . 5 (𝑝 = suc 𝑛 → (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω))
10 bianir 1057 . . . . 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 579 . . 3 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝 ∈ ω)
13 df-suc 6375 . . . . . 6 suc 𝑛 = (𝑛 ∪ {𝑛})
1413eqeq2i 2741 . . . . 5 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
15 ssun2 4173 . . . . . . 7 {𝑛} ⊆ (𝑛 ∪ {𝑛})
16 vex 3475 . . . . . . . 8 𝑛 ∈ V
1716snnz 4781 . . . . . . 7 {𝑛} ≠ ∅
18 ssn0 4401 . . . . . . 7 (({𝑛} ⊆ (𝑛 ∪ {𝑛}) ∧ {𝑛} ≠ ∅) → (𝑛 ∪ {𝑛}) ≠ ∅)
1915, 17, 18mp2an 691 . . . . . 6 (𝑛 ∪ {𝑛}) ≠ ∅
20 neeq1 3000 . . . . . 6 (𝑝 = (𝑛 ∪ {𝑛}) → (𝑝 ≠ ∅ ↔ (𝑛 ∪ {𝑛}) ≠ ∅))
2119, 20mpbiri 258 . . . . 5 (𝑝 = (𝑛 ∪ {𝑛}) → 𝑝 ≠ ∅)
2214, 21sylbi 216 . . . 4 (𝑝 = suc 𝑛𝑝 ≠ ∅)
2322adantl 481 . . 3 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝 ≠ ∅)
246eleq2i 2821 . . . 4 (𝑝𝐷𝑝 ∈ (ω ∖ {∅}))
25 eldifsn 4791 . . . 4 (𝑝 ∈ (ω ∖ {∅}) ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
2624, 25bitri 275 . . 3 (𝑝𝐷 ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
2712, 23, 26sylanbrc 582 . 2 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝𝐷)
284, 5, 27syl2anc 583 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2937  cdif 3944  cun 3945  wss 3947  c0 4323  {csn 4629  suc csuc 6371   Fn wfn 6543  ωcom 7870  w-bnj17 34317   FrSe w-bnj15 34323
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 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-om 7871  df-bnj17 34318
This theorem is referenced by:  bnj910  34579
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