Theorem exmidontri2or 7244

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 7226	. . 3 ⊢ (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
2		onelss 4389	. . . . . . . 8 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
3	2	adantl 277	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
4		orc 712	. . . . . . 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 3211	. . . . . . . 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 4389	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
10	9	adantr 276	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
11		olc 711	. . . . . . 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 1304	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
14	13	ralimdva 2544	. . . 4 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
15	14	ralimia 2538	. . 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 4573	. . . 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 4202	. 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 708  DECID wdc 834   ∨ w3o 977   = wceq 1353   ∈ wcel 2148  ∀wral 2455   ⊆ wss 3131  ∅c0 3424  {csn 3594  EXMIDwem 4196  Oncon0 4365

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-exmid 4197  df-iord 4368  df-on 4370  df-suc 4373

This theorem is referenced by:  onntri52  7245