Theorem onntri35 7368

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 7369), (3) implies (5) (onntri35 7368), (5) implies (1) (onntri51 7371), (2) implies (4) (onntri24 7373), (4) implies (5) (onntri45 7372), and (5) implies (2) (onntri52 7375).

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 7370 and exmidontri2or 7374, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7376 and (4) by onntri2or 7377.

(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 7357	. . . . 5 ⊢ 𝒫 1o ∈ On
2	1	onsuci 4572	. . . 4 ⊢ suc 𝒫 1o ∈ On
3		3on 6526	. . . 4 ⊢ 3o ∈ On
4		eleq1 2269	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 ∈ 𝑦 ↔ suc 𝒫 1o ∈ 𝑦))
5		eqeq1 2213	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑥 = 𝑦 ↔ suc 𝒫 1o = 𝑦))
6		eleq2 2270	. . . . . . . 8 ⊢ (𝑥 = suc 𝒫 1o → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝒫 1o))
7	4, 5, 6	3orbi123d 1324	. . . . . . 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 669	. . . . . 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 669	. . . . 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 2270	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o ∈ 𝑦 ↔ suc 𝒫 1o ∈ 3o))
11		eqeq2 2216	. . . . . . . 8 ⊢ (𝑦 = 3o → (suc 𝒫 1o = 𝑦 ↔ suc 𝒫 1o = 3o))
12		eleq1 2269	. . . . . . . 8 ⊢ (𝑦 = 3o → (𝑦 ∈ suc 𝒫 1o ↔ 3o ∈ suc 𝒫 1o))
13	10, 11, 12	3orbi123d 1324	. . . . . . 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 669	. . . . . 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 669	. . . . 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 2894	. . . 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 996	. . 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 678	. 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 7364	. . . 4 ⊢ ¬ suc 𝒫 1o ∈ 3o
21	20	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o ∈ 3o)
22		2on 6524	. . . . . . 7 ⊢ 2o ∈ On
23		suc11 4614	. . . . . . 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 6517	. . . . . . 7 ⊢ 3o = suc 2o
26	25	eqeq2i 2217	. . . . . 6 ⊢ (suc 𝒫 1o = 3o ↔ suc 𝒫 1o = suc 2o)
27		exmidpweq 7021	. . . . . 6 ⊢ (EXMID ↔ 𝒫 1o = 2o)
28	24, 26, 27	3bitr4ri 213	. . . . 5 ⊢ (EXMID ↔ suc 𝒫 1o = 3o)
29	28	notbii 670	. . . 4 ⊢ (¬ EXMID ↔ ¬ suc 𝒫 1o = 3o)
30	29	biimpi 120	. . 3 ⊢ (¬ EXMID → ¬ suc 𝒫 1o = 3o)
31		3nelsucpw1 7365	. . . 4 ⊢ ¬ 3o ∈ suc 𝒫 1o
32	31	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ 3o ∈ suc 𝒫 1o)
33	21, 30, 32	3jca 1180	. 2 ⊢ (¬ EXMID → (¬ suc 𝒫 1o ∈ 3o ∧ ¬ suc 𝒫 1o = 3o ∧ ¬ 3o ∈ suc 𝒫 1o))
34	19, 33	nsyl 629	1 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ w3o 980   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485  𝒫 cpw 3621  EXMIDwem 4246  Oncon0 4418  suc csuc 4420  1_oc1o 6508  2_oc2o 6509  3_oc3o 6510

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-uni 3857  df-int 3892  df-tr 4151  df-exmid 4247  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-1o 6515  df-2o 6516  df-3o 6517

This theorem is referenced by:  onntri3or  7376