Theorem ordtri2orexmid 4193

Description: Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)

Hypothesis

Ref	Expression
ordtri2orexmid.1	⊢ ∀x ∈ On ∀y ∈ On (x ∈ y ∨ y ⊆ x)

Assertion

Ref	Expression
ordtri2orexmid	⊢ (φ ∨ ¬ φ)

Distinct variable group:   φ,x,y

Proof of Theorem ordtri2orexmid

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtri2orexmid.1	. . . 4 ⊢ ∀x ∈ On ∀y ∈ On (x ∈ y ∨ y ⊆ x)
2		ordtriexmidlem 4190	. . . . 5 ⊢ {z ∈ {∅} ∣ φ} ∈ On
3		suc0 4095	. . . . . 6 ⊢ suc ∅ = {∅}
4		0elon 4076	. . . . . . 7 ⊢ ∅ ∈ On
5	4	onsuci 4189	. . . . . 6 ⊢ suc ∅ ∈ On
6	3, 5	eqeltrri 2094	. . . . 5 ⊢ {∅} ∈ On
7		eleq1 2083	. . . . . . 7 ⊢ (x = {z ∈ {∅} ∣ φ} → (x ∈ y ↔ {z ∈ {∅} ∣ φ} ∈ y))
8		sseq2 2943	. . . . . . 7 ⊢ (x = {z ∈ {∅} ∣ φ} → (y ⊆ x ↔ y ⊆ {z ∈ {∅} ∣ φ}))
9	7, 8	orbi12d 694	. . . . . 6 ⊢ (x = {z ∈ {∅} ∣ φ} → ((x ∈ y ∨ y ⊆ x) ↔ ({z ∈ {∅} ∣ φ} ∈ y ∨ y ⊆ {z ∈ {∅} ∣ φ})))
10		eleq2 2084	. . . . . . 7 ⊢ (y = {∅} → ({z ∈ {∅} ∣ φ} ∈ y ↔ {z ∈ {∅} ∣ φ} ∈ {∅}))
11		sseq1 2942	. . . . . . 7 ⊢ (y = {∅} → (y ⊆ {z ∈ {∅} ∣ φ} ↔ {∅} ⊆ {z ∈ {∅} ∣ φ}))
12	10, 11	orbi12d 694	. . . . . 6 ⊢ (y = {∅} → (({z ∈ {∅} ∣ φ} ∈ y ∨ y ⊆ {z ∈ {∅} ∣ φ}) ↔ ({z ∈ {∅} ∣ φ} ∈ {∅} ∨ {∅} ⊆ {z ∈ {∅} ∣ φ})))
13	9, 12	rspc2va 2639	. . . . 5 ⊢ ((({z ∈ {∅} ∣ φ} ∈ On ∧ {∅} ∈ On) ∧ ∀x ∈ On ∀y ∈ On (x ∈ y ∨ y ⊆ x)) → ({z ∈ {∅} ∣ φ} ∈ {∅} ∨ {∅} ⊆ {z ∈ {∅} ∣ φ}))
14	2, 6, 13	mpanl12 414	. . . 4 ⊢ (∀x ∈ On ∀y ∈ On (x ∈ y ∨ y ⊆ x) → ({z ∈ {∅} ∣ φ} ∈ {∅} ∨ {∅} ⊆ {z ∈ {∅} ∣ φ}))
15	1, 14	ax-mp 7	. . 3 ⊢ ({z ∈ {∅} ∣ φ} ∈ {∅} ∨ {∅} ⊆ {z ∈ {∅} ∣ φ})
16		elsni 3373	. . . . 5 ⊢ ({z ∈ {∅} ∣ φ} ∈ {∅} → {z ∈ {∅} ∣ φ} = ∅)
17		ordtriexmidlem2 4191	. . . . 5 ⊢ ({z ∈ {∅} ∣ φ} = ∅ → ¬ φ)
18	16, 17	syl 14	. . . 4 ⊢ ({z ∈ {∅} ∣ φ} ∈ {∅} → ¬ φ)
19		snssg 3473	. . . . . 6 ⊢ (∅ ∈ On → (∅ ∈ {z ∈ {∅} ∣ φ} ↔ {∅} ⊆ {z ∈ {∅} ∣ φ}))
20	4, 19	ax-mp 7	. . . . 5 ⊢ (∅ ∈ {z ∈ {∅} ∣ φ} ↔ {∅} ⊆ {z ∈ {∅} ∣ φ})
21		0ex 3857	. . . . . . . 8 ⊢ ∅ ∈ V
22	21	snid 3376	. . . . . . 7 ⊢ ∅ ∈ {∅}
23		biidd 161	. . . . . . . 8 ⊢ (z = ∅ → (φ ↔ φ))
24	23	elrab3 2675	. . . . . . 7 ⊢ (∅ ∈ {∅} → (∅ ∈ {z ∈ {∅} ∣ φ} ↔ φ))
25	22, 24	ax-mp 7	. . . . . 6 ⊢ (∅ ∈ {z ∈ {∅} ∣ φ} ↔ φ)
26	25	biimpi 113	. . . . 5 ⊢ (∅ ∈ {z ∈ {∅} ∣ φ} → φ)
27	20, 26	sylbir 125	. . . 4 ⊢ ({∅} ⊆ {z ∈ {∅} ∣ φ} → φ)
28	18, 27	orim12i 663	. . 3 ⊢ (({z ∈ {∅} ∣ φ} ∈ {∅} ∨ {∅} ⊆ {z ∈ {∅} ∣ φ}) → (¬ φ ∨ φ))
29	15, 28	ax-mp 7	. 2 ⊢ (¬ φ ∨ φ)
30		orcom 634	. 2 ⊢ ((¬ φ ∨ φ) ↔ (φ ∨ ¬ φ))
31	29, 30	mpbi 133	1 ⊢ (φ ∨ ¬ φ)

Syntax hints:  ¬ wn 3   ↔ wb 98   ∨ wo 616   = wceq 1228   ∈ wcel 1375  ∀wral 2283  {crab 2287   ⊆ wss 2893  ∅c0 3200  {csn 3349  Oncon0 4047  suc csuc 4049

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-nul 3856  ax-pow 3900  ax-pr 3917  ax-un 4118

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-rab 2292  df-v 2536  df-dif 2896  df-un 2898  df-in 2900  df-ss 2907  df-nul 3201  df-pw 3335  df-sn 3355  df-pr 3356  df-uni 3554  df-tr 3828  df-iord 4050  df-on 4052  df-suc 4055

This theorem is referenced by: (None)