Theorem onntri35 7151

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 7152), (3) implies (5) (onntri35 7151), (5) implies (1) (onntri51 7154), (2) implies (4) (onntri24 7156), (4) implies (5) (onntri45 7155), and (5) implies (2) (onntri52 7158).

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 7153 and exmidontri2or 7157, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7159 and (4) by onntri2or 7160.

(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 7140	. . . . 5 ⊢ 𝒫 1o ∈ On
2	1	onsuci 4469	. . . 4 ⊢ suc 𝒫 1o ∈ On
3		3on 6364	. . . 4 ⊢ 3o ∈ On
4		eleq1 2217	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 ∈ 𝑦 ↔ suc 𝒫 1o ∈ 𝑦))
5		eqeq1 2161	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 = 𝑦 ↔ suc 𝒫 1o = 𝑦))
6		eleq2 2218	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝒫 1o))
7	4, 5, 6	3orbi123d 1290	. . . . . . 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 657	. . . . . 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 657	. . . . 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 2218	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o ∈ 𝑦 ↔ suc 𝒫 1o ∈ 3o))
11		eqeq2 2164	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o = 𝑦 ↔ suc 𝒫 1o = 3o))
12		eleq1 2217	. . . . . . . 8 ⊢ (𝑦 = 3o → (𝑦 ∈ suc 𝒫 1o ↔ 3o ∈ suc 𝒫 1o))
13	10, 11, 12	3orbi123d 1290	. . . . . . 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 657	. . . . . 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 657	. . . . 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 2826	. . . 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 423	. . 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 978	. . 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 666	. 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 7147	. . . 4 ⊢ ¬ suc 𝒫 1o ∈ 3o
21	20	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o ∈ 3o)
22		2on 6362	. . . . . . 7 ⊢ 2o ∈ On
23		suc11 4511	. . . . . . 7 ⊢ ((𝒫 1o ∈ On ∧ 2o ∈ On) → (suc 𝒫 1o = suc 2o ↔ 𝒫 1o = 2o))
24	1, 22, 23	mp2an 423	. . . . . 6 ⊢ (suc 𝒫 1o = suc 2o ↔ 𝒫 1o = 2o)
25		df-3o 6355	. . . . . . 7 ⊢ 3o = suc 2o
26	25	eqeq2i 2165	. . . . . 6 ⊢ (suc 𝒫 1o = 3o ↔ suc 𝒫 1o = suc 2o)
27		exmidpweq 6843	. . . . . 6 ⊢ (EXMID ↔ 𝒫 1o = 2o)
28	24, 26, 27	3bitr4ri 212	. . . . 5 ⊢ (EXMID ↔ suc 𝒫 1o = 3o)
29	28	notbii 658	. . . 4 ⊢ (¬ EXMID ↔ ¬ suc 𝒫 1o = 3o)
30	29	biimpi 119	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o = 3o)
31		3nelsucpw1 7148	. . . 4 ⊢ ¬ 3o ∈ suc 𝒫 1o
32	31	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ 3o ∈ suc 𝒫 1o)
33	21, 30, 32	3jca 1162	. 2 ⊢ (¬ EXMID → (¬ suc 𝒫 1o ∈ 3o ∧ ¬ suc 𝒫 1o = 3o ∧ ¬ 3o ∈ suc 𝒫 1o))
34	19, 33	nsyl 618	1 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ w3o 962   ∧ w3a 963   = wceq 1332   ∈ wcel 2125  ∀wral 2432  𝒫 cpw 3539  EXMIDwem 4150  Oncon0 4318  suc csuc 4320  1_oc1o 6346  2_oc2o 6347  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-3or 964  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:  onntri3or  7159