Theorem nnwetri 7107

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 4710	. . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴)
2		ordwe 4674	. . 3 ⊢ (Ord 𝐴 → E We 𝐴)
3	1, 2	syl 14	. 2 ⊢ (𝐴 ∈ ω → E We 𝐴)
4		simprl 531	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
5		simpl 109	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐴 ∈ ω)
6		elnn 4704	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑥 ∈ ω)
7	4, 5, 6	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ω)
8		simprr 533	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
9		elnn 4704	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑦 ∈ ω)
10	8, 5, 9	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ω)
11		nntri3or 6660	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
12		epel 4389	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
13		biid 171	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
14		epel 4389	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
15	12, 13, 14	3orbi123i 1215	. . . . 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 2614	. 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 1003   ∈ wcel 2202  ∀wral 2510   class class class wbr 4088   E cep 4384   We wwe 4427  Ord word 4459  ωcom 4688

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-eprel 4386  df-frfor 4428  df-frind 4429  df-wetr 4431  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689

This theorem is referenced by: (None)