Theorem onntri45 7155

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 7140	. . . . 5 ⊢ 𝒫 1o ∈ On
2	1	onsuci 4469	. . . 4 ⊢ suc 𝒫 1o ∈ On
3		3on 6364	. . . 4 ⊢ 3o ∈ On
4		sseq1 3147	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 ⊆ 𝑦 ↔ suc 𝒫 1o ⊆ 𝑦))
5		sseq2 3148	. . . . . . . 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 3148	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o ⊆ 𝑦 ↔ suc 𝒫 1o ⊆ 3o))
10		sseq1 3147	. . . . . . . 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 2826	. . . 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 7149	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o ⊆ 3o)
19		3nsssucpw1 7150	. . 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 1332   ∈ wcel 2125  ∀wral 2432   ⊆ wss 3098  𝒫 cpw 3539  EXMIDwem 4150  Oncon0 4318  suc csuc 4320  1_oc1o 6346  3_oc3o 6348

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-v 2711  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-uni 3769  df-int 3804  df-tr 4059  df-exmid 4151  df-iord 4321  df-on 4323  df-suc 4326  df-iom 4544  df-1o 6353  df-2o 6354  df-3o 6355

This theorem is referenced by:  onntri2or  7160