Theorem onntri35 7418

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 7419), (3) implies (5) (onntri35 7418), (5) implies (1) (onntri51 7421), (2) implies (4) (onntri24 7423), (4) implies (5) (onntri45 7422), and (5) implies (2) (onntri52 7425).

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 7420 and exmidontri2or 7424, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7426 and (4) by onntri2or 7427.

(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 7407	. . . . 5 ⊢ 𝒫 1o ∈ On
2	1	onsuci 4607	. . . 4 ⊢ suc 𝒫 1o ∈ On
3		3on 6571	. . . 4 ⊢ 3o ∈ On
4		eleq1 2292	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 ∈ 𝑦 ↔ suc 𝒫 1o ∈ 𝑦))
5		eqeq1 2236	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 = 𝑦 ↔ suc 𝒫 1o = 𝑦))
6		eleq2 2293	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝒫 1o))
7	4, 5, 6	3orbi123d 1345	. . . . . . 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 671	. . . . . 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 671	. . . . 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 2293	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o ∈ 𝑦 ↔ suc 𝒫 1o ∈ 3o))
11		eqeq2 2239	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o = 𝑦 ↔ suc 𝒫 1o = 3o))
12		eleq1 2292	. . . . . . . 8 ⊢ (𝑦 = 3o → (𝑦 ∈ suc 𝒫 1o ↔ 3o ∈ suc 𝒫 1o))
13	10, 11, 12	3orbi123d 1345	. . . . . . 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 671	. . . . . 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 671	. . . . 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 2920	. . . 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 1017	. . 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 680	. 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 7414	. . . 4 ⊢ ¬ suc 𝒫 1o ∈ 3o
21	20	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o ∈ 3o)
22		2on 6569	. . . . . . 7 ⊢ 2o ∈ On
23		suc11 4649	. . . . . . 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 6562	. . . . . . 7 ⊢ 3o = suc 2o
26	25	eqeq2i 2240	. . . . . 6 ⊢ (suc 𝒫 1o = 3o ↔ suc 𝒫 1o = suc 2o)
27		exmidpweq 7067	. . . . . 6 ⊢ (EXMID ↔ 𝒫 1o = 2o)
28	24, 26, 27	3bitr4ri 213	. . . . 5 ⊢ (EXMID ↔ suc 𝒫 1o = 3o)
29	28	notbii 672	. . . 4 ⊢ (¬ EXMID ↔ ¬ suc 𝒫 1o = 3o)
30	29	biimpi 120	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o = 3o)
31		3nelsucpw1 7415	. . . 4 ⊢ ¬ 3o ∈ suc 𝒫 1o
32	31	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ 3o ∈ suc 𝒫 1o)
33	21, 30, 32	3jca 1201	. 2 ⊢ (¬ EXMID → (¬ suc 𝒫 1o ∈ 3o ∧ ¬ suc 𝒫 1o = 3o ∧ ¬ 3o ∈ suc 𝒫 1o))
34	19, 33	nsyl 631	1 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ w3o 1001   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  𝒫 cpw 3649  EXMIDwem 4277  Oncon0 4453  suc csuc 4455  1_oc1o 6553  2_oc2o 6554  3_oc3o 6555

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-int 3923  df-tr 4182  df-exmid 4278  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-1o 6560  df-2o 6561  df-3o 6562

This theorem is referenced by:  onntri3or  7426