Theorem exmidundif 4167

Description: Excluded middle is equivalent to every subset having a complement. That is, the union of a subset and its relative complement being the whole set. Although special cases such as undifss 3474 and undifdcss 6867 are provable, the full statement is equivalent to excluded middle as shown here. (Contributed by Jim Kingdon, 18-Jun-2022.)

Assertion

Ref	Expression
exmidundif	⊢ (EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))

Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidundif

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		undifss 3474	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) ⊆ 𝑦)
2	1	biimpi 119	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) ⊆ 𝑦)
3	2	adantl 275	. . . . . 6 ⊢ ((EXMID ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∪ (𝑦 ∖ 𝑥)) ⊆ 𝑦)
4		elun1 3274	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
5	4	adantl 275	. . . . . . . . . 10 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
6		simplr 520	. . . . . . . . . . . 12 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑦)
7		simpr 109	. . . . . . . . . . . 12 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → ¬ 𝑧 ∈ 𝑥)
8	6, 7	eldifd 3112	. . . . . . . . . . 11 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝑦 ∖ 𝑥))
9		elun2 3275	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑦 ∖ 𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
10	8, 9	syl 14	. . . . . . . . . 10 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
11		exmidexmid 4157	. . . . . . . . . . . 12 ⊢ (EXMID → DECID 𝑧 ∈ 𝑥)
12		exmiddc 822	. . . . . . . . . . . 12 ⊢ (DECID 𝑧 ∈ 𝑥 → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥))
13	11, 12	syl 14	. . . . . . . . . . 11 ⊢ (EXMID → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥))
14	13	adantr 274	. . . . . . . . . 10 ⊢ ((EXMID ∧ 𝑧 ∈ 𝑦) → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥))
15	5, 10, 14	mpjaodan 788	. . . . . . . . 9 ⊢ ((EXMID ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
16	15	ex 114	. . . . . . . 8 ⊢ (EXMID → (𝑧 ∈ 𝑦 → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥))))
17	16	ssrdv 3134	. . . . . . 7 ⊢ (EXMID → 𝑦 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥)))
18	17	adantr 274	. . . . . 6 ⊢ ((EXMID ∧ 𝑥 ⊆ 𝑦) → 𝑦 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥)))
19	3, 18	eqssd 3145	. . . . 5 ⊢ ((EXMID ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)
20	19	ex 114	. . . 4 ⊢ (EXMID → (𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
21		ssun1 3270	. . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥))
22		sseq2 3152	. . . . 5 ⊢ ((𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦 → (𝑥 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥)) ↔ 𝑥 ⊆ 𝑦))
23	21, 22	mpbii 147	. . . 4 ⊢ ((𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦 → 𝑥 ⊆ 𝑦)
24	20, 23	impbid1 141	. . 3 ⊢ (EXMID → (𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
25	24	alrimivv 1855	. 2 ⊢ (EXMID → ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
26		vex 2715	. . . . . 6 ⊢ 𝑧 ∈ V
27		p0ex 4149	. . . . . 6 ⊢ {∅} ∈ V
28		sseq12 3153	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → (𝑥 ⊆ 𝑦 ↔ 𝑧 ⊆ {∅}))
29		simpl 108	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → 𝑥 = 𝑧)
30		simpr 109	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → 𝑦 = {∅})
31	30, 29	difeq12d 3226	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → (𝑦 ∖ 𝑥) = ({∅} ∖ 𝑧))
32	29, 31	uneq12d 3262	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → (𝑥 ∪ (𝑦 ∖ 𝑥)) = (𝑧 ∪ ({∅} ∖ 𝑧)))
33	32, 30	eqeq12d 2172	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → ((𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦 ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
34	28, 33	bibi12d 234	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → ((𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) ↔ (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
35	34	spc2gv 2803	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ {∅} ∈ V) → (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
36	26, 27, 35	mp2an 423	. . . . 5 ⊢ (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
37		0ex 4091	. . . . . . . 8 ⊢ ∅ ∈ V
38	37	snid 3591	. . . . . . 7 ⊢ ∅ ∈ {∅}
39		eleq2 2221	. . . . . . 7 ⊢ ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ ∅ ∈ {∅}))
40	38, 39	mpbiri 167	. . . . . 6 ⊢ ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → ∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)))
41		eldifn 3230	. . . . . . . 8 ⊢ (∅ ∈ ({∅} ∖ 𝑧) → ¬ ∅ ∈ 𝑧)
42	41	orim2i 751	. . . . . . 7 ⊢ ((∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)) → (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
43		elun 3248	. . . . . . 7 ⊢ (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ (∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)))
44		df-dc 821	. . . . . . 7 ⊢ (DECID ∅ ∈ 𝑧 ↔ (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
45	42, 43, 44	3imtr4i 200	. . . . . 6 ⊢ (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) → DECID ∅ ∈ 𝑧)
46	40, 45	syl 14	. . . . 5 ⊢ ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → DECID ∅ ∈ 𝑧)
47	36, 46	syl6bi 162	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
48	47	alrimiv 1854	. . 3 ⊢ (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
49		df-exmid 4156	. . 3 ⊢ (EXMID ↔ ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
50	48, 49	sylibr 133	. 2 ⊢ (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → EXMID)
51	25, 50	impbii 125	1 ⊢ (EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 820  ∀wal 1333   = wceq 1335   ∈ wcel 2128  Vcvv 2712   ∖ cdif 3099   ∪ cun 3100   ⊆ wss 3102  ∅c0 3394  {csn 3560  EXMIDwem 4155

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-exmid 4156

This theorem is referenced by: (None)