Theorem onsucsssucexmid 4498

Description: The converse of onsucsssucr 4480 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 3222	. . . . . 6 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2		ordtriexmidlem 4490	. . . . . . 7 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3		sseq1 3160	. . . . . . . . 9 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
4		suceq 4374	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
5	4	sseq1d 3166	. . . . . . . . 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 4383	. . . . . . . . . 10 ⊢ suc ∅ = {∅}
8		0elon 4364	. . . . . . . . . . 11 ⊢ ∅ ∈ On
9	8	onsuci 4487	. . . . . . . . . 10 ⊢ suc ∅ ∈ On
10	7, 9	eqeltrri 2238	. . . . . . . . 9 ⊢ {∅} ∈ On
11		p0ex 4161	. . . . . . . . . 10 ⊢ {∅} ∈ V
12		eleq1 2227	. . . . . . . . . . . 12 ⊢ (𝑦 = {∅} → (𝑦 ∈ On ↔ {∅} ∈ On))
13	12	anbi2d 460	. . . . . . . . . . 11 ⊢ (𝑦 = {∅} → ((𝑥 ∈ On ∧ 𝑦 ∈ On) ↔ (𝑥 ∈ On ∧ {∅} ∈ On)))
14		sseq2 3161	. . . . . . . . . . . 12 ⊢ (𝑦 = {∅} → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ {∅}))
15		suceq 4374	. . . . . . . . . . . . 13 ⊢ (𝑦 = {∅} → suc 𝑦 = suc {∅})
16	15	sseq2d 3167	. . . . . . . . . . . 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 2553	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 → suc 𝑥 ⊆ suc 𝑦))
21	11, 18, 20	vtocl 2775	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ {∅} ∈ On) → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))
22	10, 21	mpan2 422	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))
23	6, 22	vtoclga 2787	. . . . . . 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 4487	. . . . . . 7 ⊢ suc {∅} ∈ On
27	26	onordi 4398	. . . . . 6 ⊢ Ord suc {∅}
28		ordelsuc 4476	. . . . . 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 4376	. . . . 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 3588	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
35		ordtriexmidlem2 4491	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
36	34, 35	syl 14	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
37		0ex 4103	. . . . 5 ⊢ ∅ ∈ V
38		biidd 171	. . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
39	37, 38	rabsnt 3645	. . . 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 1342   ∈ wcel 2135  ∀wral 2442  {crab 2446   ⊆ wss 3111  ∅c0 3404  {csn 3570  Ord word 4334  Oncon0 4335  suc csuc 4337

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-uni 3784  df-tr 4075  df-iord 4338  df-on 4340  df-suc 4343

This theorem is referenced by:  oawordriexmid  6429