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Theorem nntri3or 6357
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 2181 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
2 eqeq2 2127 . . . . 5 (𝑥 = 𝐵 → (𝐴 = 𝑥𝐴 = 𝐵))
3 eleq1 2180 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
41, 2, 33orbi123d 1274 . . . 4 (𝑥 = 𝐵 → ((𝐴𝑥𝐴 = 𝑥𝑥𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴)))
54imbi2d 229 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴𝑥𝐴 = 𝑥𝑥𝐴)) ↔ (𝐴 ∈ ω → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))))
6 eleq2 2181 . . . . 5 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
7 eqeq2 2127 . . . . 5 (𝑥 = ∅ → (𝐴 = 𝑥𝐴 = ∅))
8 eleq1 2180 . . . . 5 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ∈ 𝐴))
96, 7, 83orbi123d 1274 . . . 4 (𝑥 = ∅ → ((𝐴𝑥𝐴 = 𝑥𝑥𝐴) ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
10 eleq2 2181 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
11 eqeq2 2127 . . . . 5 (𝑥 = 𝑦 → (𝐴 = 𝑥𝐴 = 𝑦))
12 eleq1 2180 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1310, 11, 123orbi123d 1274 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑥𝐴 = 𝑥𝑥𝐴) ↔ (𝐴𝑦𝐴 = 𝑦𝑦𝐴)))
14 eleq2 2181 . . . . 5 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
15 eqeq2 2127 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 = 𝑥𝐴 = suc 𝑦))
16 eleq1 2180 . . . . 5 (𝑥 = suc 𝑦 → (𝑥𝐴 ↔ suc 𝑦𝐴))
1714, 15, 163orbi123d 1274 . . . 4 (𝑥 = suc 𝑦 → ((𝐴𝑥𝐴 = 𝑥𝑥𝐴) ↔ (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
18 0elnn 4502 . . . . 5 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
19 olc 685 . . . . . 6 ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (𝐴 ∈ ∅ ∨ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
20 3orass 950 . . . . . 6 ((𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ (𝐴 ∈ ∅ ∨ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
2119, 20sylibr 133 . . . . 5 ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴))
2218, 21syl 14 . . . 4 (𝐴 ∈ ω → (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴))
23 df-3or 948 . . . . . 6 ((𝐴𝑦𝐴 = 𝑦𝑦𝐴) ↔ ((𝐴𝑦𝐴 = 𝑦) ∨ 𝑦𝐴))
24 elex 2671 . . . . . . . 8 (𝑦 ∈ ω → 𝑦 ∈ V)
25 elsuc2g 4297 . . . . . . . . 9 (𝑦 ∈ V → (𝐴 ∈ suc 𝑦 ↔ (𝐴𝑦𝐴 = 𝑦)))
26 3mix1 1135 . . . . . . . . 9 (𝐴 ∈ suc 𝑦 → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴))
2725, 26syl6bir 163 . . . . . . . 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 6355 . . . . . . . . 9 (𝐴 ∈ ω → (𝑦𝐴 ↔ suc 𝑦 ∈ suc 𝐴))
30 elsuci 4295 . . . . . . . . 9 (suc 𝑦 ∈ suc 𝐴 → (suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴))
3129, 30syl6bi 162 . . . . . . . 8 (𝐴 ∈ ω → (𝑦𝐴 → (suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴)))
32 eqcom 2119 . . . . . . . . . . . . 13 (suc 𝑦 = 𝐴𝐴 = suc 𝑦)
3332orbi2i 736 . . . . . . . . . . . 12 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) ↔ (suc 𝑦𝐴𝐴 = suc 𝑦))
3433biimpi 119 . . . . . . . . . . 11 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) → (suc 𝑦𝐴𝐴 = suc 𝑦))
3534orcomd 703 . . . . . . . . . 10 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 = suc 𝑦 ∨ suc 𝑦𝐴))
3635olcd 708 . . . . . . . . 9 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 ∈ suc 𝑦 ∨ (𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
37 3orass 950 . . . . . . . . 9 ((𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴) ↔ (𝐴 ∈ suc 𝑦 ∨ (𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
3836, 37sylibr 133 . . . . . . . 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 693 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴𝑦𝐴 = 𝑦) ∨ 𝑦𝐴) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
4123, 40syl5bi 151 . . . . 5 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴𝑦𝐴 = 𝑦𝑦𝐴) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
4241ex 114 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴𝑦𝐴 = 𝑦𝑦𝐴) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴))))
439, 13, 17, 22, 42finds2 4485 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴𝑥𝐴 = 𝑥𝑥𝐴)))
445, 43vtoclga 2726 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴𝐵𝐴 = 𝐵𝐵𝐴)))
4544impcom 124 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 682  w3o 946   = wceq 1316  wcel 1465  Vcvv 2660  c0 3333  suc csuc 4257  ωcom 4474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475
This theorem is referenced by:  nntri2  6358  nntri1  6360  nntri3  6361  nntri2or2  6362  nndceq  6363  nndcel  6364  nnsseleq  6365  nntr2  6367  nnawordex  6392  nnwetri  6772  ltsopi  7096  pitri3or  7098  frec2uzlt2d  10145  ennnfonelemk  11840  ennnfonelemex  11854  nninfalllemn  13129
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