Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj970 Structured version   Visualization version   GIF version

Theorem bnj970 32207
Description: Technical lemma for bnj69 32270. 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 32063 . . . 4 (𝜒𝑛𝐷)
323ad2ant1 1127 . . 3 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑛𝐷)
43adantl 484 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑛𝐷)
5 simpr3 1190 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛)
6 bnj970.10 . . . . 5 𝐷 = (ω ∖ {∅})
76bnj923 32027 . . . 4 (𝑛𝐷𝑛 ∈ ω)
8 peano2 7594 . . . . 5 (𝑛 ∈ ω → suc 𝑛 ∈ ω)
9 eleq1 2898 . . . . 5 (𝑝 = suc 𝑛 → (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω))
10 bianir 1052 . . . . 5 ((suc 𝑛 ∈ ω ∧ (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω)) → 𝑝 ∈ ω)
118, 9, 10syl2an 597 . . . 4 ((𝑛 ∈ ω ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω)
127, 11sylan 582 . . 3 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝 ∈ ω)
13 df-suc 6190 . . . . . 6 suc 𝑛 = (𝑛 ∪ {𝑛})
1413eqeq2i 2832 . . . . 5 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
15 ssun2 4147 . . . . . . 7 {𝑛} ⊆ (𝑛 ∪ {𝑛})
16 vex 3496 . . . . . . . 8 𝑛 ∈ V
1716snnz 4703 . . . . . . 7 {𝑛} ≠ ∅
18 ssn0 4352 . . . . . . 7 (({𝑛} ⊆ (𝑛 ∪ {𝑛}) ∧ {𝑛} ≠ ∅) → (𝑛 ∪ {𝑛}) ≠ ∅)
1915, 17, 18mp2an 690 . . . . . 6 (𝑛 ∪ {𝑛}) ≠ ∅
20 neeq1 3076 . . . . . 6 (𝑝 = (𝑛 ∪ {𝑛}) → (𝑝 ≠ ∅ ↔ (𝑛 ∪ {𝑛}) ≠ ∅))
2119, 20mpbiri 260 . . . . 5 (𝑝 = (𝑛 ∪ {𝑛}) → 𝑝 ≠ ∅)
2214, 21sylbi 219 . . . 4 (𝑝 = suc 𝑛𝑝 ≠ ∅)
2322adantl 484 . . 3 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝 ≠ ∅)
246eleq2i 2902 . . . 4 (𝑝𝐷𝑝 ∈ (ω ∖ {∅}))
25 eldifsn 4711 . . . 4 (𝑝 ∈ (ω ∖ {∅}) ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
2624, 25bitri 277 . . 3 (𝑝𝐷 ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
2712, 23, 26sylanbrc 585 . 2 ((𝑛𝐷𝑝 = suc 𝑛) → 𝑝𝐷)
284, 5, 27syl2anc 586 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wne 3014  cdif 3931  cun 3932  wss 3934  c0 4289  {csn 4559  suc csuc 6186   Fn wfn 6343  ωcom 7572  w-bnj17 31944   FrSe w-bnj15 31950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-om 7573  df-bnj17 31945
This theorem is referenced by:  bnj910  32208
  Copyright terms: Public domain W3C validator