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Theorem bnj970 30271
 Description: Technical lemma for bnj69 30332. 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 30128 . . . 4 (𝜒𝑛𝐷)
323ad2ant1 1075 . . 3 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑛𝐷)
43adantl 481 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑛𝐷)
5 simpr3 1062 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛)
6 bnj970.10 . . . . 5 𝐷 = (ω ∖ {∅})
76bnj923 30092 . . . 4 (𝑛𝐷𝑛 ∈ ω)
8 peano2 6978 . . . . 5 (𝑛 ∈ ω → suc 𝑛 ∈ ω)
9 eleq1 2676 . . . . 5 (𝑝 = suc 𝑛 → (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω))
10 bianir 1001 . . . . 5 ((suc 𝑛 ∈ ω ∧ (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω)) → 𝑝 ∈ ω)
118, 9, 10syl2an 493 . . . 4 ((𝑛 ∈ ω ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω)
127, 11sylan 487 . . 3 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝 ∈ ω)
13 df-suc 5646 . . . . . 6 suc 𝑛 = (𝑛 ∪ {𝑛})
1413eqeq2i 2622 . . . . 5 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
15 ssun2 3739 . . . . . . 7 {𝑛} ⊆ (𝑛 ∪ {𝑛})
16 vex 3176 . . . . . . . 8 𝑛 ∈ V
1716snnz 4252 . . . . . . 7 {𝑛} ≠ ∅
18 ssn0 3928 . . . . . . 7 (({𝑛} ⊆ (𝑛 ∪ {𝑛}) ∧ {𝑛} ≠ ∅) → (𝑛 ∪ {𝑛}) ≠ ∅)
1915, 17, 18mp2an 704 . . . . . 6 (𝑛 ∪ {𝑛}) ≠ ∅
20 neeq1 2844 . . . . . 6 (𝑝 = (𝑛 ∪ {𝑛}) → (𝑝 ≠ ∅ ↔ (𝑛 ∪ {𝑛}) ≠ ∅))
2119, 20mpbiri 247 . . . . 5 (𝑝 = (𝑛 ∪ {𝑛}) → 𝑝 ≠ ∅)
2214, 21sylbi 206 . . . 4 (𝑝 = suc 𝑛𝑝 ≠ ∅)
2322adantl 481 . . 3 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝 ≠ ∅)
246eleq2i 2680 . . . 4 (𝑝𝐷𝑝 ∈ (ω ∖ {∅}))
25 eldifsn 4260 . . . 4 (𝑝 ∈ (ω ∖ {∅}) ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
2624, 25bitri 263 . . 3 (𝑝𝐷 ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
2712, 23, 26sylanbrc 695 . 2 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝𝐷)
284, 5, 27syl2anc 691 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  suc csuc 5642   Fn wfn 5799  ωcom 6957   ∧ w-bnj17 30005   FrSe w-bnj15 30011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958  df-bnj17 30006 This theorem is referenced by:  bnj910  30272
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