Theorem onntri45 7170

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 7155	. . . . 5 ⊢ 𝒫 1o ∈ On
2	1	onsuci 4474	. . . 4 ⊢ suc 𝒫 1o ∈ On
3		3on 6371	. . . 4 ⊢ 3o ∈ On
4		sseq1 3151	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 ⊆ 𝑦 ↔ suc 𝒫 1o ⊆ 𝑦))
5		sseq2 3152	. . . . . . . 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 3152	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o ⊆ 𝑦 ↔ suc 𝒫 1o ⊆ 3o))
10		sseq1 3151	. . . . . . . 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 2829	. . . 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 7164	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o ⊆ 3o)
19		3nsssucpw1 7165	. . 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 1335   ∈ wcel 2128  ∀wral 2435   ⊆ wss 3102  𝒫 cpw 3543  EXMIDwem 4155  Oncon0 4323  suc csuc 4325  1_oc1o 6353  3_oc3o 6355

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-tr 4063  df-exmid 4156  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-1o 6360  df-2o 6361  df-3o 6362

This theorem is referenced by:  onntri2or  7175