Theorem onntri35 7238

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 7239), (3) implies (5) (onntri35 7238), (5) implies (1) (onntri51 7241), (2) implies (4) (onntri24 7243), (4) implies (5) (onntri45 7242), and (5) implies (2) (onntri52 7245).

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 7240 and exmidontri2or 7244, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7246 and (4) by onntri2or 7247.

(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 7227	. . . . 5 ⊢ 𝒫 1o ∈ On
2	1	onsuci 4517	. . . 4 ⊢ suc 𝒫 1o ∈ On
3		3on 6430	. . . 4 ⊢ 3o ∈ On
4		eleq1 2240	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 ∈ 𝑦 ↔ suc 𝒫 1o ∈ 𝑦))
5		eqeq1 2184	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 = 𝑦 ↔ suc 𝒫 1o = 𝑦))
6		eleq2 2241	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝒫 1o))
7	4, 5, 6	3orbi123d 1311	. . . . . . 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 667	. . . . . 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 667	. . . . 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 2241	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o ∈ 𝑦 ↔ suc 𝒫 1o ∈ 3o))
11		eqeq2 2187	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o = 𝑦 ↔ suc 𝒫 1o = 3o))
12		eleq1 2240	. . . . . . . 8 ⊢ (𝑦 = 3o → (𝑦 ∈ suc 𝒫 1o ↔ 3o ∈ suc 𝒫 1o))
13	10, 11, 12	3orbi123d 1311	. . . . . . 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 667	. . . . . 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 667	. . . . 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 2856	. . . 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 993	. . 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 676	. 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 7234	. . . 4 ⊢ ¬ suc 𝒫 1o ∈ 3o
21	20	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o ∈ 3o)
22		2on 6428	. . . . . . 7 ⊢ 2o ∈ On
23		suc11 4559	. . . . . . 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 6421	. . . . . . 7 ⊢ 3o = suc 2o
26	25	eqeq2i 2188	. . . . . 6 ⊢ (suc 𝒫 1o = 3o ↔ suc 𝒫 1o = suc 2o)
27		exmidpweq 6911	. . . . . 6 ⊢ (EXMID ↔ 𝒫 1o = 2o)
28	24, 26, 27	3bitr4ri 213	. . . . 5 ⊢ (EXMID ↔ suc 𝒫 1o = 3o)
29	28	notbii 668	. . . 4 ⊢ (¬ EXMID ↔ ¬ suc 𝒫 1o = 3o)
30	29	biimpi 120	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o = 3o)
31		3nelsucpw1 7235	. . . 4 ⊢ ¬ 3o ∈ suc 𝒫 1o
32	31	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ 3o ∈ suc 𝒫 1o)
33	21, 30, 32	3jca 1177	. 2 ⊢ (¬ EXMID → (¬ suc 𝒫 1o ∈ 3o ∧ ¬ suc 𝒫 1o = 3o ∧ ¬ 3o ∈ suc 𝒫 1o))
34	19, 33	nsyl 628	1 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ w3o 977   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455  𝒫 cpw 3577  EXMIDwem 4196  Oncon0 4365  suc csuc 4367  1_oc1o 6412  2_oc2o 6413  3_oc3o 6414

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-tr 4104  df-exmid 4197  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-1o 6419  df-2o 6420  df-3o 6421

This theorem is referenced by:  onntri3or  7246