Theorem onntri45 7564

Description: Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)

Assertion

Ref	Expression
onntri45	⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ EXMID)

Distinct variable group:   𝑥,𝑦

Proof of Theorem onntri45

Step	Hyp	Ref	Expression
1		pw1on 7549	. . . . 5 ⊢ 𝒫 1o ∈ On
2	1	onsuci 4643	. . . 4 ⊢ suc 𝒫 1o ∈ On
3		3on 6671	. . . 4 ⊢ 3o ∈ On
4		sseq1 3265	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 ⊆ 𝑦 ↔ suc 𝒫 1o ⊆ 𝑦))
5		sseq2 3266	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ suc 𝒫 1o))
6	4, 5	orbi12d 801	. . . . . . 7 ⊢ (𝑥 = suc 𝒫 1o → ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o)))
7	6	notbid 673	. . . . . 6 ⊢ (𝑥 = suc 𝒫 1o → (¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ¬ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o)))
8	7	notbid 673	. . . . 5 ⊢ (𝑥 = suc 𝒫 1o → (¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ¬ ¬ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o)))
9		sseq2 3266	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o ⊆ 𝑦 ↔ suc 𝒫 1o ⊆ 3o))
10		sseq1 3265	. . . . . . . 8 ⊢ (𝑦 = 3o → (𝑦 ⊆ suc 𝒫 1o ↔ 3o ⊆ suc 𝒫 1o))
11	9, 10	orbi12d 801	. . . . . . 7 ⊢ (𝑦 = 3o → ((suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o) ↔ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
12	11	notbid 673	. . . . . 6 ⊢ (𝑦 = 3o → (¬ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o) ↔ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
13	12	notbid 673	. . . . 5 ⊢ (𝑦 = 3o → (¬ ¬ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o) ↔ ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
14	8, 13	rspc2v 2937	. . . 4 ⊢ ((suc 𝒫 1o ∈ On ∧ 3o ∈ On) → (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
15	2, 3, 14	mp2an 426	. . 3 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o))
16		ioran 760	. . 3 ⊢ (¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o) ↔ (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
17	15, 16	sylnib 683	. 2 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
18		sucpw1nss3 7558	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o ⊆ 3o)
19		3nsssucpw1 7559	. . 3 ⊢ (¬ EXMID → ¬ 3o ⊆ suc 𝒫 1o)
20	18, 19	jca 306	. 2 ⊢ (¬ EXMID → (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
21	17, 20	nsyl 633	1 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398   ∈ wcel 2205  ∀wral 2522   ⊆ wss 3214  𝒫 cpw 3674  EXMIDwem 4312  Oncon0 4489  suc csuc 4491  1_oc1o 6653  3_oc3o 6655

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-int 3955  df-tr 4214  df-exmid 4313  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-1o 6660  df-2o 6661  df-3o 6662

This theorem is referenced by:  onntri2or  7569