Theorem exmidontri2or 7504

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 7483	. . 3 ⊢ (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
2		onelss 4490	. . . . . . . 8 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
3	2	adantl 277	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
4		orc 720	. . . . . . 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 3282	. . . . . . . 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 4490	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
10	9	adantr 276	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
11		olc 719	. . . . . . 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 1341	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
14	13	ralimdva 2600	. . . 4 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
15	14	ralimia 2594	. . 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 4676	. . . 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 4296	. 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 716  DECID wdc 842   ∨ w3o 1004   = wceq 1398   ∈ wcel 2202  ∀wral 2511   ⊆ wss 3201  ∅c0 3496  {csn 3673  EXMIDwem 4290  Oncon0 4466

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-tr 4193  df-exmid 4291  df-iord 4469  df-on 4471  df-suc 4474

This theorem is referenced by:  onntri52  7505