Theorem exmidundif 4239

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 3531 and undifdcss 6984 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 3531	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) ⊆ 𝑦)
2	1	biimpi 120	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) ⊆ 𝑦)
3	2	adantl 277	. . . . . 6 ⊢ ((EXMID ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∪ (𝑦 ∖ 𝑥)) ⊆ 𝑦)
4		elun1 3330	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
5	4	adantl 277	. . . . . . . . . 10 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
6		simplr 528	. . . . . . . . . . . 12 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑦)
7		simpr 110	. . . . . . . . . . . 12 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → ¬ 𝑧 ∈ 𝑥)
8	6, 7	eldifd 3167	. . . . . . . . . . 11 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝑦 ∖ 𝑥))
9		elun2 3331	. . . . . . . . . . 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 4229	. . . . . . . . . . . 12 ⊢ (EXMID → DECID 𝑧 ∈ 𝑥)
12		exmiddc 837	. . . . . . . . . . . 12 ⊢ (DECID 𝑧 ∈ 𝑥 → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥))
13	11, 12	syl 14	. . . . . . . . . . 11 ⊢ (EXMID → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥))
14	13	adantr 276	. . . . . . . . . 10 ⊢ ((EXMID ∧ 𝑧 ∈ 𝑦) → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥))
15	5, 10, 14	mpjaodan 799	. . . . . . . . 9 ⊢ ((EXMID ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
16	15	ex 115	. . . . . . . 8 ⊢ (EXMID → (𝑧 ∈ 𝑦 → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥))))
17	16	ssrdv 3189	. . . . . . 7 ⊢ (EXMID → 𝑦 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥)))
18	17	adantr 276	. . . . . 6 ⊢ ((EXMID ∧ 𝑥 ⊆ 𝑦) → 𝑦 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥)))
19	3, 18	eqssd 3200	. . . . 5 ⊢ ((EXMID ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)
20	19	ex 115	. . . 4 ⊢ (EXMID → (𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
21		ssun1 3326	. . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥))
22		sseq2 3207	. . . . 5 ⊢ ((𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦 → (𝑥 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥)) ↔ 𝑥 ⊆ 𝑦))
23	21, 22	mpbii 148	. . . 4 ⊢ ((𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦 → 𝑥 ⊆ 𝑦)
24	20, 23	impbid1 142	. . 3 ⊢ (EXMID → (𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
25	24	alrimivv 1889	. 2 ⊢ (EXMID → ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
26		vex 2766	. . . . . 6 ⊢ 𝑧 ∈ V
27		p0ex 4221	. . . . . 6 ⊢ {∅} ∈ V
28		sseq12 3208	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → (𝑥 ⊆ 𝑦 ↔ 𝑧 ⊆ {∅}))
29		simpl 109	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → 𝑥 = 𝑧)
30		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → 𝑦 = {∅})
31	30, 29	difeq12d 3282	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → (𝑦 ∖ 𝑥) = ({∅} ∖ 𝑧))
32	29, 31	uneq12d 3318	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → (𝑥 ∪ (𝑦 ∖ 𝑥)) = (𝑧 ∪ ({∅} ∖ 𝑧)))
33	32, 30	eqeq12d 2211	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → ((𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦 ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
34	28, 33	bibi12d 235	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → ((𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) ↔ (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
35	34	spc2gv 2855	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ {∅} ∈ V) → (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
36	26, 27, 35	mp2an 426	. . . . 5 ⊢ (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
37		0ex 4160	. . . . . . . 8 ⊢ ∅ ∈ V
38	37	snid 3653	. . . . . . 7 ⊢ ∅ ∈ {∅}
39		eleq2 2260	. . . . . . 7 ⊢ ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ ∅ ∈ {∅}))
40	38, 39	mpbiri 168	. . . . . 6 ⊢ ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → ∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)))
41		eldifn 3286	. . . . . . . 8 ⊢ (∅ ∈ ({∅} ∖ 𝑧) → ¬ ∅ ∈ 𝑧)
42	41	orim2i 762	. . . . . . 7 ⊢ ((∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)) → (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
43		elun 3304	. . . . . . 7 ⊢ (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ (∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)))
44		df-dc 836	. . . . . . 7 ⊢ (DECID ∅ ∈ 𝑧 ↔ (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
45	42, 43, 44	3imtr4i 201	. . . . . 6 ⊢ (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) → DECID ∅ ∈ 𝑧)
46	40, 45	syl 14	. . . . 5 ⊢ ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → DECID ∅ ∈ 𝑧)
47	36, 46	biimtrdi 163	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
48	47	alrimiv 1888	. . 3 ⊢ (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
49		df-exmid 4228	. . 3 ⊢ (EXMID ↔ ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
50	48, 49	sylibr 134	. 2 ⊢ (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → EXMID)
51	25, 50	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835  ∀wal 1362   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∖ cdif 3154   ∪ cun 3155   ⊆ wss 3157  ∅c0 3450  {csn 3622  EXMIDwem 4227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-exmid 4228

This theorem is referenced by: (None)