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

Theorem nnwetri 6354
 Description: A natural number is well-ordered by E. More specifically, this order both satisfies We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
Assertion
Ref Expression
nnwetri (𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem nnwetri
StepHypRef Expression
1 nnord 4334 . . 3 (𝐴 ∈ ω → Ord 𝐴)
2 ordwe 4300 . . 3 (Ord 𝐴 → E We 𝐴)
31, 2syl 14 . 2 (𝐴 ∈ ω → E We 𝐴)
4 simprl 483 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑥𝐴)
5 simpl 102 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝐴 ∈ ω)
6 elnn 4328 . . . . 5 ((𝑥𝐴𝐴 ∈ ω) → 𝑥 ∈ ω)
74, 5, 6syl2anc 391 . . . 4 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 ∈ ω)
8 simprr 484 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
9 elnn 4328 . . . . 5 ((𝑦𝐴𝐴 ∈ ω) → 𝑦 ∈ ω)
108, 5, 9syl2anc 391 . . . 4 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑦 ∈ ω)
11 nntri3or 6072 . . . . 5 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
12 epel 4029 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
13 biid 160 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
14 epel 4029 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
1512, 13, 143orbi123i 1094 . . . . 5 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
1611, 15sylibr 137 . . . 4 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
177, 10, 16syl2anc 391 . . 3 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
1817ralrimivva 2401 . 2 (𝐴 ∈ ω → ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
193, 18jca 290 1 (𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ w3o 884   ∈ wcel 1393  ∀wral 2306   class class class wbr 3764   E cep 4024   We wwe 4067  Ord word 4099  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-setind 4262  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-eu 1903  df-mo 1904  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-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-tr 3855  df-eprel 4026  df-frfor 4068  df-frind 4069  df-wetr 4071  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator