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Theorem nnwetri 6908
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 4607 . . 3 (𝐴 ∈ ω → Ord 𝐴)
2 ordwe 4571 . . 3 (Ord 𝐴 → E We 𝐴)
31, 2syl 14 . 2 (𝐴 ∈ ω → E We 𝐴)
4 simprl 529 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑥𝐴)
5 simpl 109 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝐴 ∈ ω)
6 elnn 4601 . . . . 5 ((𝑥𝐴𝐴 ∈ ω) → 𝑥 ∈ ω)
74, 5, 6syl2anc 411 . . . 4 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 ∈ ω)
8 simprr 531 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
9 elnn 4601 . . . . 5 ((𝑦𝐴𝐴 ∈ ω) → 𝑦 ∈ ω)
108, 5, 9syl2anc 411 . . . 4 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑦 ∈ ω)
11 nntri3or 6487 . . . . 5 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
12 epel 4288 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
13 biid 171 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
14 epel 4288 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
1512, 13, 143orbi123i 1189 . . . . 5 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
1611, 15sylibr 134 . . . 4 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
177, 10, 16syl2anc 411 . . 3 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
1817ralrimivva 2559 . 2 (𝐴 ∈ ω → ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
193, 18jca 306 1 (𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3o 977  wcel 2148  wral 2455   class class class wbr 4000   E cep 4283   We wwe 4326  Ord word 4358  ωcom 4585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-tr 4099  df-eprel 4285  df-frfor 4327  df-frind 4328  df-wetr 4330  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586
This theorem is referenced by: (None)
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