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Theorem nnwetri 6436
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 4360 . . 3 (𝐴 ∈ ω → Ord 𝐴)
2 ordwe 4326 . . 3 (Ord 𝐴 → E We 𝐴)
31, 2syl 14 . 2 (𝐴 ∈ ω → E We 𝐴)
4 simprl 498 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑥𝐴)
5 simpl 107 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝐴 ∈ ω)
6 elnn 4354 . . . . 5 ((𝑥𝐴𝐴 ∈ ω) → 𝑥 ∈ ω)
74, 5, 6syl2anc 403 . . . 4 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 ∈ ω)
8 simprr 499 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
9 elnn 4354 . . . . 5 ((𝑦𝐴𝐴 ∈ ω) → 𝑦 ∈ ω)
108, 5, 9syl2anc 403 . . . 4 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑦 ∈ ω)
11 nntri3or 6137 . . . . 5 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
12 epel 4055 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
13 biid 169 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
14 epel 4055 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
1512, 13, 143orbi123i 1129 . . . . 5 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
1611, 15sylibr 132 . . . 4 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
177, 10, 16syl2anc 403 . . 3 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
1817ralrimivva 2444 . 2 (𝐴 ∈ ω → ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
193, 18jca 300 1 (𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3o 919  wcel 1434  wral 2349   class class class wbr 3793   E cep 4050   We wwe 4093  Ord word 4125  ωcom 4339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-tr 3884  df-eprel 4052  df-frfor 4094  df-frind 4095  df-wetr 4097  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340
This theorem is referenced by: (None)
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