Theorem nnwetri 7028

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 4668	. . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴)
2		ordwe 4632	. . 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 4662	. . . . 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 4662	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑦 ∈ ω)
10	8, 5, 9	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ω)
11		nntri3or 6592	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
12		epel 4347	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
13		biid 171	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
14		epel 4347	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
15	12, 13, 14	3orbi123i 1192	. . . . 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 2589	. 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 980   ∈ wcel 2177  ∀wral 2485   class class class wbr 4051   E cep 4342   We wwe 4385  Ord word 4417  ωcom 4646

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-tr 4151  df-eprel 4344  df-frfor 4386  df-frind 4387  df-wetr 4389  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647

This theorem is referenced by: (None)