ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nntri3or GIF version

Theorem nntri3or 6493
Description: Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)
Assertion
Ref Expression
nntri3or ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))

Proof of Theorem nntri3or
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2241 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
2 eqeq2 2187 . . . . 5 (𝑥 = 𝐵 → (𝐴 = 𝑥𝐴 = 𝐵))
3 eleq1 2240 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
41, 2, 33orbi123d 1311 . . . 4 (𝑥 = 𝐵 → ((𝐴𝑥𝐴 = 𝑥𝑥𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴)))
54imbi2d 230 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴𝑥𝐴 = 𝑥𝑥𝐴)) ↔ (𝐴 ∈ ω → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))))
6 eleq2 2241 . . . . 5 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
7 eqeq2 2187 . . . . 5 (𝑥 = ∅ → (𝐴 = 𝑥𝐴 = ∅))
8 eleq1 2240 . . . . 5 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ∈ 𝐴))
96, 7, 83orbi123d 1311 . . . 4 (𝑥 = ∅ → ((𝐴𝑥𝐴 = 𝑥𝑥𝐴) ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
10 eleq2 2241 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
11 eqeq2 2187 . . . . 5 (𝑥 = 𝑦 → (𝐴 = 𝑥𝐴 = 𝑦))
12 eleq1 2240 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1310, 11, 123orbi123d 1311 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑥𝐴 = 𝑥𝑥𝐴) ↔ (𝐴𝑦𝐴 = 𝑦𝑦𝐴)))
14 eleq2 2241 . . . . 5 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
15 eqeq2 2187 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 = 𝑥𝐴 = suc 𝑦))
16 eleq1 2240 . . . . 5 (𝑥 = suc 𝑦 → (𝑥𝐴 ↔ suc 𝑦𝐴))
1714, 15, 163orbi123d 1311 . . . 4 (𝑥 = suc 𝑦 → ((𝐴𝑥𝐴 = 𝑥𝑥𝐴) ↔ (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
18 0elnn 4618 . . . . 5 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
19 olc 711 . . . . . 6 ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (𝐴 ∈ ∅ ∨ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
20 3orass 981 . . . . . 6 ((𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ (𝐴 ∈ ∅ ∨ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
2119, 20sylibr 134 . . . . 5 ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴))
2218, 21syl 14 . . . 4 (𝐴 ∈ ω → (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴))
23 df-3or 979 . . . . . 6 ((𝐴𝑦𝐴 = 𝑦𝑦𝐴) ↔ ((𝐴𝑦𝐴 = 𝑦) ∨ 𝑦𝐴))
24 elex 2748 . . . . . . . 8 (𝑦 ∈ ω → 𝑦 ∈ V)
25 elsuc2g 4405 . . . . . . . . 9 (𝑦 ∈ V → (𝐴 ∈ suc 𝑦 ↔ (𝐴𝑦𝐴 = 𝑦)))
26 3mix1 1166 . . . . . . . . 9 (𝐴 ∈ suc 𝑦 → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴))
2725, 26syl6bir 164 . . . . . . . 8 (𝑦 ∈ V → ((𝐴𝑦𝐴 = 𝑦) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
2824, 27syl 14 . . . . . . 7 (𝑦 ∈ ω → ((𝐴𝑦𝐴 = 𝑦) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
29 nnsucelsuc 6491 . . . . . . . . 9 (𝐴 ∈ ω → (𝑦𝐴 ↔ suc 𝑦 ∈ suc 𝐴))
30 elsuci 4403 . . . . . . . . 9 (suc 𝑦 ∈ suc 𝐴 → (suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴))
3129, 30syl6bi 163 . . . . . . . 8 (𝐴 ∈ ω → (𝑦𝐴 → (suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴)))
32 eqcom 2179 . . . . . . . . . . . . 13 (suc 𝑦 = 𝐴𝐴 = suc 𝑦)
3332orbi2i 762 . . . . . . . . . . . 12 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) ↔ (suc 𝑦𝐴𝐴 = suc 𝑦))
3433biimpi 120 . . . . . . . . . . 11 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) → (suc 𝑦𝐴𝐴 = suc 𝑦))
3534orcomd 729 . . . . . . . . . 10 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 = suc 𝑦 ∨ suc 𝑦𝐴))
3635olcd 734 . . . . . . . . 9 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 ∈ suc 𝑦 ∨ (𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
37 3orass 981 . . . . . . . . 9 ((𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴) ↔ (𝐴 ∈ suc 𝑦 ∨ (𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
3836, 37sylibr 134 . . . . . . . 8 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴))
3931, 38syl6 33 . . . . . . 7 (𝐴 ∈ ω → (𝑦𝐴 → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
4028, 39jaao 719 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴𝑦𝐴 = 𝑦) ∨ 𝑦𝐴) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
4123, 40biimtrid 152 . . . . 5 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴𝑦𝐴 = 𝑦𝑦𝐴) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
4241ex 115 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴𝑦𝐴 = 𝑦𝑦𝐴) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴))))
439, 13, 17, 22, 42finds2 4600 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴𝑥𝐴 = 𝑥𝑥𝐴)))
445, 43vtoclga 2803 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴𝐵𝐴 = 𝐵𝐵𝐴)))
4544impcom 125 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 708  w3o 977   = wceq 1353  wcel 2148  Vcvv 2737  c0 3422  suc csuc 4365  ωcom 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590
This theorem is referenced by:  nntri2  6494  nntri1  6496  nntri3  6497  nntri2or2  6498  nndceq  6499  nndcel  6500  nnsseleq  6501  nntr2  6503  nnawordex  6529  nnwetri  6914  nnnninfeq  7125  ltsopi  7318  pitri3or  7320  frec2uzlt2d  10403  ennnfonelemk  12400  ennnfonelemex  12414
  Copyright terms: Public domain W3C validator