Theorem onsucsssucexmid 4504

Description: The converse of onsucsssucr 4486 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 3227	. . . . . 6 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2		ordtriexmidlem 4496	. . . . . . 7 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3		sseq1 3165	. . . . . . . . 9 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
4		suceq 4380	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
5	4	sseq1d 3171	. . . . . . . . 9 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (suc 𝑥 ⊆ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
6	3, 5	imbi12d 233	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})))
7		suc0 4389	. . . . . . . . . 10 ⊢ suc ∅ = {∅}
8		0elon 4370	. . . . . . . . . . 11 ⊢ ∅ ∈ On
9	8	onsuci 4493	. . . . . . . . . 10 ⊢ suc ∅ ∈ On
10	7, 9	eqeltrri 2240	. . . . . . . . 9 ⊢ {∅} ∈ On
11		p0ex 4167	. . . . . . . . . 10 ⊢ {∅} ∈ V
12		eleq1 2229	. . . . . . . . . . . 12 ⊢ (𝑦 = {∅} → (𝑦 ∈ On ↔ {∅} ∈ On))
13	12	anbi2d 460	. . . . . . . . . . 11 ⊢ (𝑦 = {∅} → ((𝑥 ∈ On ∧ 𝑦 ∈ On) ↔ (𝑥 ∈ On ∧ {∅} ∈ On)))
14		sseq2 3166	. . . . . . . . . . . 12 ⊢ (𝑦 = {∅} → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ {∅}))
15		suceq 4380	. . . . . . . . . . . . 13 ⊢ (𝑦 = {∅} → suc 𝑦 = suc {∅})
16	15	sseq2d 3172	. . . . . . . . . . . 12 ⊢ (𝑦 = {∅} → (suc 𝑥 ⊆ suc 𝑦 ↔ suc 𝑥 ⊆ suc {∅}))
17	14, 16	imbi12d 233	. . . . . . . . . . 11 ⊢ (𝑦 = {∅} → ((𝑥 ⊆ 𝑦 → suc 𝑥 ⊆ suc 𝑦) ↔ (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅})))
18	13, 17	imbi12d 233	. . . . . . . . . 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 2555	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 → suc 𝑥 ⊆ suc 𝑦))
21	11, 18, 20	vtocl 2780	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ {∅} ∈ On) → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))
22	10, 21	mpan2 422	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))
23	6, 22	vtoclga 2792	. . . . . . 7 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
24	2, 23	ax-mp 5	. . . . . 6 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})
25	1, 24	ax-mp 5	. . . . 5 ⊢ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}
26	10	onsuci 4493	. . . . . . 7 ⊢ suc {∅} ∈ On
27	26	onordi 4404	. . . . . 6 ⊢ Ord suc {∅}
28		ordelsuc 4482	. . . . . 6 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ∈ On ∧ Ord suc {∅}) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
29	2, 27, 28	mp2an 423	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})
30	25, 29	mpbir 145	. . . 4 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅}
31		elsucg 4382	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅})))
32	2, 31	ax-mp 5	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅}))
33	30, 32	mpbi 144	. . 3 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅})
34		elsni 3594	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
35		ordtriexmidlem2 4497	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
36	34, 35	syl 14	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
37		0ex 4109	. . . . 5 ⊢ ∅ ∈ V
38		biidd 171	. . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
39	37, 38	rabsnt 3651	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = {∅} → 𝜑)
40	36, 39	orim12i 749	. . 3 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅}) → (¬ 𝜑 ∨ 𝜑))
41	33, 40	ax-mp 5	. 2 ⊢ (¬ 𝜑 ∨ 𝜑)
42		orcom 718	. 2 ⊢ ((¬ 𝜑 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
43	41, 42	mpbi 144	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1343   ∈ wcel 2136  ∀wral 2444  {crab 2448   ⊆ wss 3116  ∅c0 3409  {csn 3576  Ord word 4340  Oncon0 4341  suc csuc 4343

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by:  oawordriexmid  6438