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Theorem nntri3or 6072
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 2101 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
2 eqeq2 2049 . . . . 5 (𝑥 = 𝐵 → (𝐴 = 𝑥𝐴 = 𝐵))
3 eleq1 2100 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
41, 2, 33orbi123d 1206 . . . 4 (𝑥 = 𝐵 → ((𝐴𝑥𝐴 = 𝑥𝑥𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴)))
54imbi2d 219 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴𝑥𝐴 = 𝑥𝑥𝐴)) ↔ (𝐴 ∈ ω → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))))
6 eleq2 2101 . . . . 5 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
7 eqeq2 2049 . . . . 5 (𝑥 = ∅ → (𝐴 = 𝑥𝐴 = ∅))
8 eleq1 2100 . . . . 5 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ∈ 𝐴))
96, 7, 83orbi123d 1206 . . . 4 (𝑥 = ∅ → ((𝐴𝑥𝐴 = 𝑥𝑥𝐴) ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
10 eleq2 2101 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
11 eqeq2 2049 . . . . 5 (𝑥 = 𝑦 → (𝐴 = 𝑥𝐴 = 𝑦))
12 eleq1 2100 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1310, 11, 123orbi123d 1206 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑥𝐴 = 𝑥𝑥𝐴) ↔ (𝐴𝑦𝐴 = 𝑦𝑦𝐴)))
14 eleq2 2101 . . . . 5 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
15 eqeq2 2049 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 = 𝑥𝐴 = suc 𝑦))
16 eleq1 2100 . . . . 5 (𝑥 = suc 𝑦 → (𝑥𝐴 ↔ suc 𝑦𝐴))
1714, 15, 163orbi123d 1206 . . . 4 (𝑥 = suc 𝑦 → ((𝐴𝑥𝐴 = 𝑥𝑥𝐴) ↔ (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
18 0elnn 4340 . . . . 5 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
19 olc 632 . . . . . 6 ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (𝐴 ∈ ∅ ∨ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
20 3orass 888 . . . . . 6 ((𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ (𝐴 ∈ ∅ ∨ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
2119, 20sylibr 137 . . . . 5 ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴))
2218, 21syl 14 . . . 4 (𝐴 ∈ ω → (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴))
23 df-3or 886 . . . . . 6 ((𝐴𝑦𝐴 = 𝑦𝑦𝐴) ↔ ((𝐴𝑦𝐴 = 𝑦) ∨ 𝑦𝐴))
24 elex 2566 . . . . . . . 8 (𝑦 ∈ ω → 𝑦 ∈ V)
25 elsuc2g 4142 . . . . . . . . 9 (𝑦 ∈ V → (𝐴 ∈ suc 𝑦 ↔ (𝐴𝑦𝐴 = 𝑦)))
26 3mix1 1073 . . . . . . . . 9 (𝐴 ∈ suc 𝑦 → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴))
2725, 26syl6bir 153 . . . . . . . 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 6070 . . . . . . . . 9 (𝐴 ∈ ω → (𝑦𝐴 ↔ suc 𝑦 ∈ suc 𝐴))
30 elsuci 4140 . . . . . . . . 9 (suc 𝑦 ∈ suc 𝐴 → (suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴))
3129, 30syl6bi 152 . . . . . . . 8 (𝐴 ∈ ω → (𝑦𝐴 → (suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴)))
32 eqcom 2042 . . . . . . . . . . . . 13 (suc 𝑦 = 𝐴𝐴 = suc 𝑦)
3332orbi2i 679 . . . . . . . . . . . 12 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) ↔ (suc 𝑦𝐴𝐴 = suc 𝑦))
3433biimpi 113 . . . . . . . . . . 11 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) → (suc 𝑦𝐴𝐴 = suc 𝑦))
3534orcomd 648 . . . . . . . . . 10 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 = suc 𝑦 ∨ suc 𝑦𝐴))
3635olcd 653 . . . . . . . . 9 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 ∈ suc 𝑦 ∨ (𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
37 3orass 888 . . . . . . . . 9 ((𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴) ↔ (𝐴 ∈ suc 𝑦 ∨ (𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
3836, 37sylibr 137 . . . . . . . 8 ((suc 𝑦𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴))
3931, 38syl6 29 . . . . . . 7 (𝐴 ∈ ω → (𝑦𝐴 → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
4028, 39jaao 639 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴𝑦𝐴 = 𝑦) ∨ 𝑦𝐴) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
4123, 40syl5bi 141 . . . . 5 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴𝑦𝐴 = 𝑦𝑦𝐴) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴)))
4241ex 108 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴𝑦𝐴 = 𝑦𝑦𝐴) → (𝐴 ∈ suc 𝑦𝐴 = suc 𝑦 ∨ suc 𝑦𝐴))))
439, 13, 17, 22, 42finds2 4324 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴𝑥𝐴 = 𝑥𝑥𝐴)))
445, 43vtoclga 2619 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴𝐵𝐴 = 𝐵𝐵𝐴)))
4544impcom 116 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wo 629  w3o 884   = wceq 1243  wcel 1393  Vcvv 2557  c0 3224  suc csuc 4102  ωcom 4313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314
This theorem is referenced by:  nntri2  6073  nntri1  6074  nntri3  6075  nntri2or2  6076  nndceq  6077  nndcel  6078  nnsseleq  6079  nnawordex  6101  nnwetri  6354  ltsopi  6416  pitri3or  6418  frec2uzlt2d  9164
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