Theorem exmidontri2or 7326

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 7308	. . 3 ⊢ (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
2		onelss 4423	. . . . . . . 8 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
3	2	adantl 277	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
4		orc 713	. . . . . . 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 3238	. . . . . . . 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 4423	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
10	9	adantr 276	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
11		olc 712	. . . . . . 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 1315	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
14	13	ralimdva 2564	. . . 4 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
15	14	ralimia 2558	. . 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 4609	. . . 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 4234	. 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 709  DECID wdc 835   ∨ w3o 979   = wceq 1364   ∈ wcel 2167  ∀wral 2475   ⊆ wss 3157  ∅c0 3451  {csn 3623  EXMIDwem 4228  Oncon0 4399

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-uni 3841  df-tr 4133  df-exmid 4229  df-iord 4402  df-on 4404  df-suc 4407

This theorem is referenced by:  onntri52  7327