Theorem onntri35 7547

Description: Double negated ordinal trichotomy.

There are five equivalent statements: (1) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥), (2) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥), (3) ∀𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥), (4) ∀𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥), and (5) ¬ ¬ EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7548), (3) implies (5) (onntri35 7547), (5) implies (1) (onntri51 7550), (2) implies (4) (onntri24 7552), (4) implies (5) (onntri45 7551), and (5) implies (2) (onntri52 7554).

Another way of stating this is that EXMID is equivalent to trichotomy, either the 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 or the 𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥 form, as shown in exmidontri 7549 and exmidontri2or 7553, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7555 and (4) by onntri2or 7556.

(Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)

Assertion

Ref	Expression
onntri35	⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID)

Distinct variable group:   𝑥,𝑦

Proof of Theorem onntri35

Step	Hyp	Ref	Expression
1		pw1on 7536	. . . . 5 ⊢ 𝒫 1o ∈ On
2	1	onsuci 4638	. . . 4 ⊢ suc 𝒫 1o ∈ On
3		3on 6658	. . . 4 ⊢ 3o ∈ On
4		eleq1 2295	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 ∈ 𝑦 ↔ suc 𝒫 1o ∈ 𝑦))
5		eqeq1 2239	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 = 𝑦 ↔ suc 𝒫 1o = 𝑦))
6		eleq2 2296	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝒫 1o))
7	4, 5, 6	3orbi123d 1348	. . . . . . 7 ⊢ (𝑥 = suc 𝒫 1o → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (suc 𝒫 1o ∈ 𝑦 ∨ suc 𝒫 1o = 𝑦 ∨ 𝑦 ∈ suc 𝒫 1o)))
8	7	notbid 673	. . . . . 6 ⊢ (𝑥 = suc 𝒫 1o → (¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ¬ (suc 𝒫 1o ∈ 𝑦 ∨ suc 𝒫 1o = 𝑦 ∨ 𝑦 ∈ suc 𝒫 1o)))
9	8	notbid 673	. . . . 5 ⊢ (𝑥 = suc 𝒫 1o → (¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ¬ ¬ (suc 𝒫 1o ∈ 𝑦 ∨ suc 𝒫 1o = 𝑦 ∨ 𝑦 ∈ suc 𝒫 1o)))
10		eleq2 2296	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o ∈ 𝑦 ↔ suc 𝒫 1o ∈ 3o))
11		eqeq2 2242	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o = 𝑦 ↔ suc 𝒫 1o = 3o))
12		eleq1 2295	. . . . . . . 8 ⊢ (𝑦 = 3o → (𝑦 ∈ suc 𝒫 1o ↔ 3o ∈ suc 𝒫 1o))
13	10, 11, 12	3orbi123d 1348	. . . . . . 7 ⊢ (𝑦 = 3o → ((suc 𝒫 1o ∈ 𝑦 ∨ suc 𝒫 1o = 𝑦 ∨ 𝑦 ∈ suc 𝒫 1o) ↔ (suc 𝒫 1o ∈ 3o ∨ suc 𝒫 1o = 3o ∨ 3o ∈ suc 𝒫 1o)))
14	13	notbid 673	. . . . . 6 ⊢ (𝑦 = 3o → (¬ (suc 𝒫 1o ∈ 𝑦 ∨ suc 𝒫 1o = 𝑦 ∨ 𝑦 ∈ suc 𝒫 1o) ↔ ¬ (suc 𝒫 1o ∈ 3o ∨ suc 𝒫 1o = 3o ∨ 3o ∈ suc 𝒫 1o)))
15	14	notbid 673	. . . . 5 ⊢ (𝑦 = 3o → (¬ ¬ (suc 𝒫 1o ∈ 𝑦 ∨ suc 𝒫 1o = 𝑦 ∨ 𝑦 ∈ suc 𝒫 1o) ↔ ¬ ¬ (suc 𝒫 1o ∈ 3o ∨ suc 𝒫 1o = 3o ∨ 3o ∈ suc 𝒫 1o)))
16	9, 15	rspc2v 2934	. . . 4 ⊢ ((suc 𝒫 1o ∈ On ∧ 3o ∈ On) → (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ (suc 𝒫 1o ∈ 3o ∨ suc 𝒫 1o = 3o ∨ 3o ∈ suc 𝒫 1o)))
17	2, 3, 16	mp2an 426	. . 3 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ (suc 𝒫 1o ∈ 3o ∨ suc 𝒫 1o = 3o ∨ 3o ∈ suc 𝒫 1o))
18		3ioran 1020	. . 3 ⊢ (¬ (suc 𝒫 1o ∈ 3o ∨ suc 𝒫 1o = 3o ∨ 3o ∈ suc 𝒫 1o) ↔ (¬ suc 𝒫 1o ∈ 3o ∧ ¬ suc 𝒫 1o = 3o ∧ ¬ 3o ∈ suc 𝒫 1o))
19	17, 18	sylnib 683	. 2 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ (¬ suc 𝒫 1o ∈ 3o ∧ ¬ suc 𝒫 1o = 3o ∧ ¬ 3o ∈ suc 𝒫 1o))
20		sucpw1nel3 7543	. . . 4 ⊢ ¬ suc 𝒫 1o ∈ 3o
21	20	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o ∈ 3o)
22		2on 6656	. . . . . . 7 ⊢ 2o ∈ On
23		suc11 4680	. . . . . . 7 ⊢ ((𝒫 1o ∈ On ∧ 2o ∈ On) → (suc 𝒫 1o = suc 2o ↔ 𝒫 1o = 2o))
24	1, 22, 23	mp2an 426	. . . . . 6 ⊢ (suc 𝒫 1o = suc 2o ↔ 𝒫 1o = 2o)
25		df-3o 6649	. . . . . . 7 ⊢ 3o = suc 2o
26	25	eqeq2i 2243	. . . . . 6 ⊢ (suc 𝒫 1o = 3o ↔ suc 𝒫 1o = suc 2o)
27		exmidpweq 7169	. . . . . 6 ⊢ (EXMID ↔ 𝒫 1o = 2o)
28	24, 26, 27	3bitr4ri 213	. . . . 5 ⊢ (EXMID ↔ suc 𝒫 1o = 3o)
29	28	notbii 674	. . . 4 ⊢ (¬ EXMID ↔ ¬ suc 𝒫 1o = 3o)
30	29	biimpi 120	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o = 3o)
31		3nelsucpw1 7544	. . . 4 ⊢ ¬ 3o ∈ suc 𝒫 1o
32	31	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ 3o ∈ suc 𝒫 1o)
33	21, 30, 32	3jca 1204	. 2 ⊢ (¬ EXMID → (¬ suc 𝒫 1o ∈ 3o ∧ ¬ suc 𝒫 1o = 3o ∧ ¬ 3o ∈ suc 𝒫 1o))
34	19, 33	nsyl 633	1 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ w3o 1004   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ∀wral 2520  𝒫 cpw 3669  EXMIDwem 4307  Oncon0 4484  suc csuc 4486  1_oc1o 6640  2_oc2o 6641  3_oc3o 6642

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-tr 4209  df-exmid 4308  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-1o 6647  df-2o 6648  df-3o 6649

This theorem is referenced by:  onntri3or  7555