Theorem exmidontri2or 7199

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 7181	. . 3 ⊢ (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
2		onelss 4365	. . . . . . . 8 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
3	2	adantl 275	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
4		orc 702	. . . . . . 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 3196	. . . . . . . 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 4365	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
10	9	adantr 274	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
11		olc 701	. . . . . . 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 1294	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
14	13	ralimdva 2533	. . . 4 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
15	14	ralimia 2527	. . 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 4549	. . . 4 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → DECID 𝑧 = {∅})
18	17	adantr 274	. . 3 ⊢ ((∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ∧ 𝑧 ⊆ {∅}) → DECID 𝑧 = {∅})
19	18	exmid1dc 4179	. 2 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → EXMID)
20	16, 19	impbii 125	1 ⊢ (EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 824   ∨ w3o 967   = wceq 1343   ∈ wcel 2136  ∀wral 2444   ⊆ wss 3116  ∅c0 3409  {csn 3576  EXMIDwem 4173  Oncon0 4341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-exmid 4174  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by:  onntri52  7200