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Theorem nnwetri 6849
 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 4565 . . 3 (𝐴 ∈ ω → Ord 𝐴)
2 ordwe 4529 . . 3 (Ord 𝐴 → E We 𝐴)
31, 2syl 14 . 2 (𝐴 ∈ ω → E We 𝐴)
4 simprl 521 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑥𝐴)
5 simpl 108 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝐴 ∈ ω)
6 elnn 4559 . . . . 5 ((𝑥𝐴𝐴 ∈ ω) → 𝑥 ∈ ω)
74, 5, 6syl2anc 409 . . . 4 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 ∈ ω)
8 simprr 522 . . . . 5 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
9 elnn 4559 . . . . 5 ((𝑦𝐴𝐴 ∈ ω) → 𝑦 ∈ ω)
108, 5, 9syl2anc 409 . . . 4 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → 𝑦 ∈ ω)
11 nntri3or 6429 . . . . 5 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
12 epel 4247 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
13 biid 170 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
14 epel 4247 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
1512, 13, 143orbi123i 1172 . . . . 5 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
1611, 15sylibr 133 . . . 4 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
177, 10, 16syl2anc 409 . . 3 ((𝐴 ∈ ω ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
1817ralrimivva 2536 . 2 (𝐴 ∈ ω → ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
193, 18jca 304 1 (𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∨ w3o 962   ∈ wcel 2125  ∀wral 2432   class class class wbr 3961   E cep 4242   We wwe 4285  Ord word 4317  ωcom 4543 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541 This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-br 3962  df-opab 4022  df-tr 4059  df-eprel 4244  df-frfor 4286  df-frind 4287  df-wetr 4289  df-iord 4321  df-on 4323  df-suc 4326  df-iom 4544 This theorem is referenced by: (None)
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