Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj168 Structured version   Visualization version   GIF version

Theorem bnj168 33741
Description: First-order logic and set theory. Revised to remove dependence on ax-reg 9587. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by NM, 21-Dec-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj168.1 𝐷 = (ω ∖ {∅})
Assertion
Ref Expression
bnj168 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚𝐷 𝑛 = suc 𝑚)
Distinct variable group:   𝑚,𝑛
Allowed substitution hints:   𝐷(𝑚,𝑛)

Proof of Theorem bnj168
StepHypRef Expression
1 bnj168.1 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
21bnj158 33740 . . . . . . . . 9 (𝑛𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
32anim2i 618 . . . . . . . 8 ((𝑛 ≠ 1o𝑛𝐷) → (𝑛 ≠ 1o ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4 r19.42v 3191 . . . . . . . 8 (∃𝑚 ∈ ω (𝑛 ≠ 1o𝑛 = suc 𝑚) ↔ (𝑛 ≠ 1o ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
53, 4sylibr 233 . . . . . . 7 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 ≠ 1o𝑛 = suc 𝑚))
6 neeq1 3004 . . . . . . . . . . 11 (𝑛 = suc 𝑚 → (𝑛 ≠ 1o ↔ suc 𝑚 ≠ 1o))
76biimpac 480 . . . . . . . . . 10 ((𝑛 ≠ 1o𝑛 = suc 𝑚) → suc 𝑚 ≠ 1o)
8 df-1o 8466 . . . . . . . . . . . . 13 1o = suc ∅
98eqeq2i 2746 . . . . . . . . . . . 12 (suc 𝑚 = 1o ↔ suc 𝑚 = suc ∅)
10 nnon 7861 . . . . . . . . . . . . 13 (𝑚 ∈ ω → 𝑚 ∈ On)
11 0elon 6419 . . . . . . . . . . . . 13 ∅ ∈ On
12 suc11 6472 . . . . . . . . . . . . 13 ((𝑚 ∈ On ∧ ∅ ∈ On) → (suc 𝑚 = suc ∅ ↔ 𝑚 = ∅))
1310, 11, 12sylancl 587 . . . . . . . . . . . 12 (𝑚 ∈ ω → (suc 𝑚 = suc ∅ ↔ 𝑚 = ∅))
149, 13bitr2id 284 . . . . . . . . . . 11 (𝑚 ∈ ω → (𝑚 = ∅ ↔ suc 𝑚 = 1o))
1514necon3bid 2986 . . . . . . . . . 10 (𝑚 ∈ ω → (𝑚 ≠ ∅ ↔ suc 𝑚 ≠ 1o))
167, 15imbitrrid 245 . . . . . . . . 9 (𝑚 ∈ ω → ((𝑛 ≠ 1o𝑛 = suc 𝑚) → 𝑚 ≠ ∅))
1716ancld 552 . . . . . . . 8 (𝑚 ∈ ω → ((𝑛 ≠ 1o𝑛 = suc 𝑚) → ((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅)))
1817reximia 3082 . . . . . . 7 (∃𝑚 ∈ ω (𝑛 ≠ 1o𝑛 = suc 𝑚) → ∃𝑚 ∈ ω ((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅))
195, 18syl 17 . . . . . 6 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚 ∈ ω ((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅))
20 anass 470 . . . . . . 7 (((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅) ↔ (𝑛 ≠ 1o ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2120rexbii 3095 . . . . . 6 (∃𝑚 ∈ ω ((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅) ↔ ∃𝑚 ∈ ω (𝑛 ≠ 1o ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2219, 21sylib 217 . . . . 5 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 ≠ 1o ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
23 simpr 486 . . . . 5 ((𝑛 ≠ 1o ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑛 = suc 𝑚𝑚 ≠ ∅))
2422, 23bnj31 33730 . . . 4 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 = suc 𝑚𝑚 ≠ ∅))
25 df-rex 3072 . . . 4 (∃𝑚 ∈ ω (𝑛 = suc 𝑚𝑚 ≠ ∅) ↔ ∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2624, 25sylib 217 . . 3 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
27 simpr 486 . . . . . . 7 ((𝑛 = suc 𝑚𝑚 ≠ ∅) → 𝑚 ≠ ∅)
2827anim2i 618 . . . . . 6 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
291eleq2i 2826 . . . . . . 7 (𝑚𝐷𝑚 ∈ (ω ∖ {∅}))
30 eldifsn 4791 . . . . . . 7 (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
3129, 30bitr2i 276 . . . . . 6 ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) ↔ 𝑚𝐷)
3228, 31sylib 217 . . . . 5 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → 𝑚𝐷)
33 simprl 770 . . . . 5 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → 𝑛 = suc 𝑚)
3432, 33jca 513 . . . 4 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑚𝐷𝑛 = suc 𝑚))
3534eximi 1838 . . 3 (∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
3626, 35syl 17 . 2 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
37 df-rex 3072 . 2 (∃𝑚𝐷 𝑛 = suc 𝑚 ↔ ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
3836, 37sylibr 233 1 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚𝐷 𝑛 = suc 𝑚)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2941  wrex 3071  cdif 3946  c0 4323  {csn 4629  Oncon0 6365  suc csuc 6367  ωcom 7855  1oc1o 8459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-om 7856  df-1o 8466
This theorem is referenced by:  bnj600  33930
  Copyright terms: Public domain W3C validator