Theorem onntri45 7197

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 7182	. . . . 5 ⊢ 𝒫 1o ∈ On
2	1	onsuci 4493	. . . 4 ⊢ suc 𝒫 1o ∈ On
3		3on 6395	. . . 4 ⊢ 3o ∈ On
4		sseq1 3165	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 ⊆ 𝑦 ↔ suc 𝒫 1o ⊆ 𝑦))
5		sseq2 3166	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ suc 𝒫 1o))
6	4, 5	orbi12d 783	. . . . . . 7 ⊢ (𝑥 = suc 𝒫 1o → ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o)))
7	6	notbid 657	. . . . . 6 ⊢ (𝑥 = suc 𝒫 1o → (¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ¬ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o)))
8	7	notbid 657	. . . . 5 ⊢ (𝑥 = suc 𝒫 1o → (¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ¬ ¬ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o)))
9		sseq2 3166	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o ⊆ 𝑦 ↔ suc 𝒫 1o ⊆ 3o))
10		sseq1 3165	. . . . . . . 8 ⊢ (𝑦 = 3o → (𝑦 ⊆ suc 𝒫 1o ↔ 3o ⊆ suc 𝒫 1o))
11	9, 10	orbi12d 783	. . . . . . 7 ⊢ (𝑦 = 3o → ((suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o) ↔ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
12	11	notbid 657	. . . . . 6 ⊢ (𝑦 = 3o → (¬ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o) ↔ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
13	12	notbid 657	. . . . 5 ⊢ (𝑦 = 3o → (¬ ¬ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o) ↔ ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
14	8, 13	rspc2v 2843	. . . 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 423	. . 3 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o))
16		ioran 742	. . 3 ⊢ (¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o) ↔ (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
17	15, 16	sylnib 666	. 2 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
18		sucpw1nss3 7191	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o ⊆ 3o)
19		3nsssucpw1 7192	. . 3 ⊢ (¬ EXMID → ¬ 3o ⊆ suc 𝒫 1o)
20	18, 19	jca 304	. 2 ⊢ (¬ EXMID → (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
21	17, 20	nsyl 618	1 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698   = wceq 1343   ∈ wcel 2136  ∀wral 2444   ⊆ wss 3116  𝒫 cpw 3559  EXMIDwem 4173  Oncon0 4341  suc csuc 4343  1_oc1o 6377  3_oc3o 6379

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-3an 970  df-tru 1346  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-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-int 3825  df-tr 4081  df-exmid 4174  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-1o 6384  df-2o 6385  df-3o 6386

This theorem is referenced by:  onntri2or  7202