Theorem nntri3 6583

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 4589	. . . . . 6 ⊢ ¬ 𝐴 ∈ 𝐴
2		eleq2 2269	. . . . . 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 2268	. . . . . 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 6579	. 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 2176  ωcom 4638

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-tr 4143  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639

This theorem is referenced by:  frec2uzf1od  10551  nnti  15929