Theorem onsucsssucexmid 4212

Description: The converse of onsucsssucr 4200 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)

Hypothesis

Ref	Expression
onsucsssucexmid.1	⊢ ∀x ∈ On ∀y ∈ On (x ⊆ y → suc x ⊆ suc y)

Assertion

Ref	Expression
onsucsssucexmid	⊢ (φ ∨ ¬ φ)

Distinct variable groups:   φ,x   x,y

Allowed substitution hint:   φ(y)

Proof of Theorem onsucsssucexmid

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3019	. . . . . 6 ⊢ {z ∈ {∅} ∣ φ} ⊆ {∅}
2		ordtriexmidlem 4208	. . . . . . 7 ⊢ {z ∈ {∅} ∣ φ} ∈ On
3		sseq1 2960	. . . . . . . . 9 ⊢ (x = {z ∈ {∅} ∣ φ} → (x ⊆ {∅} ↔ {z ∈ {∅} ∣ φ} ⊆ {∅}))
4		suceq 4105	. . . . . . . . . 10 ⊢ (x = {z ∈ {∅} ∣ φ} → suc x = suc {z ∈ {∅} ∣ φ})
5	4	sseq1d 2966	. . . . . . . . 9 ⊢ (x = {z ∈ {∅} ∣ φ} → (suc x ⊆ suc {∅} ↔ suc {z ∈ {∅} ∣ φ} ⊆ suc {∅}))
6	3, 5	imbi12d 223	. . . . . . . 8 ⊢ (x = {z ∈ {∅} ∣ φ} → ((x ⊆ {∅} → suc x ⊆ suc {∅}) ↔ ({z ∈ {∅} ∣ φ} ⊆ {∅} → suc {z ∈ {∅} ∣ φ} ⊆ suc {∅})))
7		suc0 4114	. . . . . . . . . 10 ⊢ suc ∅ = {∅}
8		0elon 4095	. . . . . . . . . . 11 ⊢ ∅ ∈ On
9	8	onsuci 4207	. . . . . . . . . 10 ⊢ suc ∅ ∈ On
10	7, 9	eqeltrri 2108	. . . . . . . . 9 ⊢ {∅} ∈ On
11		p0ex 3930	. . . . . . . . . 10 ⊢ {∅} ∈ V
12		eleq1 2097	. . . . . . . . . . . 12 ⊢ (y = {∅} → (y ∈ On ↔ {∅} ∈ On))
13	12	anbi2d 437	. . . . . . . . . . 11 ⊢ (y = {∅} → ((x ∈ On ∧ y ∈ On) ↔ (x ∈ On ∧ {∅} ∈ On)))
14		sseq2 2961	. . . . . . . . . . . 12 ⊢ (y = {∅} → (x ⊆ y ↔ x ⊆ {∅}))
15		suceq 4105	. . . . . . . . . . . . 13 ⊢ (y = {∅} → suc y = suc {∅})
16	15	sseq2d 2967	. . . . . . . . . . . 12 ⊢ (y = {∅} → (suc x ⊆ suc y ↔ suc x ⊆ suc {∅}))
17	14, 16	imbi12d 223	. . . . . . . . . . 11 ⊢ (y = {∅} → ((x ⊆ y → suc x ⊆ suc y) ↔ (x ⊆ {∅} → suc x ⊆ suc {∅})))
18	13, 17	imbi12d 223	. . . . . . . . . 10 ⊢ (y = {∅} → (((x ∈ On ∧ y ∈ On) → (x ⊆ y → suc x ⊆ suc y)) ↔ ((x ∈ On ∧ {∅} ∈ On) → (x ⊆ {∅} → suc x ⊆ suc {∅}))))
19		onsucsssucexmid.1	. . . . . . . . . . 11 ⊢ ∀x ∈ On ∀y ∈ On (x ⊆ y → suc x ⊆ suc y)
20	19	rspec2 2402	. . . . . . . . . 10 ⊢ ((x ∈ On ∧ y ∈ On) → (x ⊆ y → suc x ⊆ suc y))
21	11, 18, 20	vtocl 2602	. . . . . . . . 9 ⊢ ((x ∈ On ∧ {∅} ∈ On) → (x ⊆ {∅} → suc x ⊆ suc {∅}))
22	10, 21	mpan2 401	. . . . . . . 8 ⊢ (x ∈ On → (x ⊆ {∅} → suc x ⊆ suc {∅}))
23	6, 22	vtoclga 2613	. . . . . . 7 ⊢ ({z ∈ {∅} ∣ φ} ∈ On → ({z ∈ {∅} ∣ φ} ⊆ {∅} → suc {z ∈ {∅} ∣ φ} ⊆ suc {∅}))
24	2, 23	ax-mp 7	. . . . . 6 ⊢ ({z ∈ {∅} ∣ φ} ⊆ {∅} → suc {z ∈ {∅} ∣ φ} ⊆ suc {∅})
25	1, 24	ax-mp 7	. . . . 5 ⊢ suc {z ∈ {∅} ∣ φ} ⊆ suc {∅}
26	10	onsuci 4207	. . . . . . 7 ⊢ suc {∅} ∈ On
27	26	onordi 4129	. . . . . 6 ⊢ Ord suc {∅}
28		ordelsuc 4197	. . . . . 6 ⊢ (({z ∈ {∅} ∣ φ} ∈ On ∧ Ord suc {∅}) → ({z ∈ {∅} ∣ φ} ∈ suc {∅} ↔ suc {z ∈ {∅} ∣ φ} ⊆ suc {∅}))
29	2, 27, 28	mp2an 402	. . . . 5 ⊢ ({z ∈ {∅} ∣ φ} ∈ suc {∅} ↔ suc {z ∈ {∅} ∣ φ} ⊆ suc {∅})
30	25, 29	mpbir 134	. . . 4 ⊢ {z ∈ {∅} ∣ φ} ∈ suc {∅}
31		elsucg 4107	. . . . 5 ⊢ ({z ∈ {∅} ∣ φ} ∈ On → ({z ∈ {∅} ∣ φ} ∈ suc {∅} ↔ ({z ∈ {∅} ∣ φ} ∈ {∅} ∨ {z ∈ {∅} ∣ φ} = {∅})))
32	2, 31	ax-mp 7	. . . 4 ⊢ ({z ∈ {∅} ∣ φ} ∈ suc {∅} ↔ ({z ∈ {∅} ∣ φ} ∈ {∅} ∨ {z ∈ {∅} ∣ φ} = {∅}))
33	30, 32	mpbi 133	. . 3 ⊢ ({z ∈ {∅} ∣ φ} ∈ {∅} ∨ {z ∈ {∅} ∣ φ} = {∅})
34		elsni 3391	. . . . 5 ⊢ ({z ∈ {∅} ∣ φ} ∈ {∅} → {z ∈ {∅} ∣ φ} = ∅)
35		ordtriexmidlem2 4209	. . . . 5 ⊢ ({z ∈ {∅} ∣ φ} = ∅ → ¬ φ)
36	34, 35	syl 14	. . . 4 ⊢ ({z ∈ {∅} ∣ φ} ∈ {∅} → ¬ φ)
37		0ex 3875	. . . . 5 ⊢ ∅ ∈ V
38		biidd 161	. . . . 5 ⊢ (z = ∅ → (φ ↔ φ))
39	37, 38	rabsnt 3436	. . . 4 ⊢ ({z ∈ {∅} ∣ φ} = {∅} → φ)
40	36, 39	orim12i 675	. . 3 ⊢ (({z ∈ {∅} ∣ φ} ∈ {∅} ∨ {z ∈ {∅} ∣ φ} = {∅}) → (¬ φ ∨ φ))
41	33, 40	ax-mp 7	. 2 ⊢ (¬ φ ∨ φ)
42		orcom 646	. 2 ⊢ ((¬ φ ∨ φ) ↔ (φ ∨ ¬ φ))
43	41, 42	mpbi 133	1 ⊢ (φ ∨ ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390  ∀wral 2300  {crab 2304   ⊆ wss 2911  ∅c0 3218  {csn 3367  Ord word 4065  Oncon0 4066  suc csuc 4068

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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074

This theorem is referenced by: (None)