Theorem nntri3 6601

Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)

Assertion

Ref	Expression
nntri3	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)))

Proof of Theorem nntri3

Step	Hyp	Ref	Expression
1		elirr 4602	. . . . . 6 ⊢ ¬ 𝐴 ∈ 𝐴
2		eleq2 2270	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵))
3	1, 2	mtbii 676	. . . . 5 ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ∈ 𝐵)
4	3	con2i 628	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 = 𝐵)
5	4	adantl 277	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) → ¬ 𝐴 = 𝐵)
6		simpl 109	. . . . 5 ⊢ ((¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴) → ¬ 𝐴 ∈ 𝐵)
7	6	con2i 628	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ¬ (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴))
8	7	adantl 277	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) → ¬ (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴))
9	5, 8	2falsed 704	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) → (𝐴 = 𝐵 ↔ (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)))
10		simpr 110	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
11		eleq1 2269	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
12	1, 11	mtbii 676	. . . . 5 ⊢ (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴)
13	3, 12	jca 306	. . . 4 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴))
14	13	adantl 277	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴))
15	10, 14	2thd 175	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 ↔ (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)))
16	12	con2i 628	. . . 4 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = 𝐵)
17	16	adantl 277	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ∈ 𝐴) → ¬ 𝐴 = 𝐵)
18		simpr 110	. . . . 5 ⊢ ((¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ 𝐴)
19	18	con2i 628	. . . 4 ⊢ (𝐵 ∈ 𝐴 → ¬ (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴))
20	19	adantl 277	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ∈ 𝐴) → ¬ (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴))
21	17, 20	2falsed 704	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ∈ 𝐴) → (𝐴 = 𝐵 ↔ (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)))
22		nntri3or 6597	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
23	9, 15, 21, 22	mpjao3dan 1320	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2177  ωcom 4651

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 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  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-uni 3860  df-int 3895  df-tr 4154  df-iord 4426  df-on 4428  df-suc 4431  df-iom 4652

This theorem is referenced by:  frec2uzf1od  10583  nnti  16099