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Theorem nnwetri 6382
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 4359 . . 3 (𝐴 ∈ ω → Ord 𝐴)
2 ordwe 4325 . . 3 (Ord 𝐴 → E We 𝐴)
31, 2syl 14 . 2 (𝐴 ∈ ω → E We 𝐴)
4 simprl 491 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑥𝐴)
5 simpl 106 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝐴 ∈ ω)
6 elnn 4353 . . . . 5 ((𝑥𝐴𝐴 ∈ ω) → 𝑥 ∈ ω)
74, 5, 6syl2anc 397 . . . 4 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 ∈ ω)
8 simprr 492 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
9 elnn 4353 . . . . 5 ((𝑦𝐴𝐴 ∈ ω) → 𝑦 ∈ ω)
108, 5, 9syl2anc 397 . . . 4 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑦 ∈ ω)
11 nntri3or 6100 . . . . 5 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
12 epel 4054 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
13 biid 164 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
14 epel 4054 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
1512, 13, 143orbi123i 1103 . . . . 5 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
1611, 15sylibr 141 . . . 4 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
177, 10, 16syl2anc 397 . . 3 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
1817ralrimivva 2416 . 2 (𝐴 ∈ ω → ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
193, 18jca 294 1 (𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3o 893  wcel 1407  wral 2321   class class class wbr 3789   E cep 4049   We wwe 4092  Ord word 4124  ωcom 4338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-3or 895  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-br 3790  df-opab 3844  df-tr 3880  df-eprel 4051  df-frfor 4093  df-frind 4094  df-wetr 4096  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339
This theorem is referenced by: (None)
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