Theorem exmidontri2or 7424

Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)

Assertion

Ref	Expression
exmidontri2or	⊢ (EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))

Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidontri2or

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidontriim 7403	. . 3 ⊢ (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
2		onelss 4477	. . . . . . . 8 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
3	2	adantl 277	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
4		orc 717	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑦 → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
5	3, 4	syl6 33	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
6		eqimss 3278	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 ⊆ 𝑦)
7	6, 4	syl 14	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
8	7	a1i 9	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 = 𝑦 → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
9		onelss 4477	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
10	9	adantr 276	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
11		olc 716	. . . . . . 7 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
12	10, 11	syl6 33	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
13	5, 8, 12	3jaod 1338	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
14	13	ralimdva 2597	. . . 4 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
15	14	ralimia 2591	. . 3 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
16	1, 15	syl 14	. 2 ⊢ (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
17		ontri2orexmidim 4663	. . . 4 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → DECID 𝑧 = {∅})
18	17	adantr 276	. . 3 ⊢ ((∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ∧ 𝑧 ⊆ {∅}) → DECID 𝑧 = {∅})
19	18	exmid1dc 4283	. 2 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → EXMID)
20	16, 19	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713  DECID wdc 839   ∨ w3o 1001   = wceq 1395   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197  ∅c0 3491  {csn 3666  EXMIDwem 4277  Oncon0 4453

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-tr 4182  df-exmid 4278  df-iord 4456  df-on 4458  df-suc 4461

This theorem is referenced by:  onntri52  7425