Theorem onntri45 7590

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 7575	. . . . 5 ⊢ 𝒫 1o ∈ On
2	1	onsuci 4658	. . . 4 ⊢ suc 𝒫 1o ∈ On
3		3on 6688	. . . 4 ⊢ 3o ∈ On
4		sseq1 3271	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 ⊆ 𝑦 ↔ suc 𝒫 1o ⊆ 𝑦))
5		sseq2 3272	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ suc 𝒫 1o))
6	4, 5	orbi12d 805	. . . . . . 7 ⊢ (𝑥 = suc 𝒫 1o → ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o)))
7	6	notbid 677	. . . . . 6 ⊢ (𝑥 = suc 𝒫 1o → (¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ¬ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o)))
8	7	notbid 677	. . . . 5 ⊢ (𝑥 = suc 𝒫 1o → (¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ¬ ¬ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o)))
9		sseq2 3272	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o ⊆ 𝑦 ↔ suc 𝒫 1o ⊆ 3o))
10		sseq1 3271	. . . . . . . 8 ⊢ (𝑦 = 3o → (𝑦 ⊆ suc 𝒫 1o ↔ 3o ⊆ suc 𝒫 1o))
11	9, 10	orbi12d 805	. . . . . . 7 ⊢ (𝑦 = 3o → ((suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o) ↔ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
12	11	notbid 677	. . . . . 6 ⊢ (𝑦 = 3o → (¬ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o) ↔ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
13	12	notbid 677	. . . . 5 ⊢ (𝑦 = 3o → (¬ ¬ (suc 𝒫 1o ⊆ 𝑦 ∨ 𝑦 ⊆ suc 𝒫 1o) ↔ ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
14	8, 13	rspc2v 2943	. . . 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 430	. . 3 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o))
16		ioran 764	. . 3 ⊢ (¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o) ↔ (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
17	15, 16	sylnib 687	. 2 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
18		sucpw1nss3 7584	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o ⊆ 3o)
19		3nsssucpw1 7585	. . 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 637	1 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  𝒫 cpw 3685  EXMIDwem 4326  Oncon0 4503  suc csuc 4505  1_oc1o 6670  3_oc3o 6672

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-tr 4225  df-exmid 4327  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-1o 6677  df-2o 6678  df-3o 6679

This theorem is referenced by:  onntri2or  7595