Theorem nnwetri 6974

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 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem nnwetri

Step	Hyp	Ref	Expression
1		nnord 4645	. . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴)
2		ordwe 4609	. . 3 ⊢ (Ord 𝐴 → E We 𝐴)
3	1, 2	syl 14	. 2 ⊢ (𝐴 ∈ ω → E We 𝐴)
4		simprl 529	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
5		simpl 109	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐴 ∈ ω)
6		elnn 4639	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑥 ∈ ω)
7	4, 5, 6	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ω)
8		simprr 531	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
9		elnn 4639	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑦 ∈ ω)
10	8, 5, 9	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ω)
11		nntri3or 6548	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
12		epel 4324	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
13		biid 171	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
14		epel 4324	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
15	12, 13, 14	3orbi123i 1191	. . . . 5 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
16	11, 15	sylibr 134	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
17	7, 10, 16	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
18	17	ralrimivva 2576	. 2 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
19	3, 18	jca 306	1 ⊢ (𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   ∨ w3o 979   ∈ wcel 2164  ∀wral 2472   class class class wbr 4030   E cep 4319   We wwe 4362  Ord word 4394  ωcom 4623

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-tr 4129  df-eprel 4321  df-frfor 4363  df-frind 4364  df-wetr 4366  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624

This theorem is referenced by: (None)