Theorem exmidundifim 4138

Description: Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4137 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidundifim

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		undifss 3448	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) ⊆ 𝑦)
2	1	biimpi 119	. . . . . 6 ⊢ (𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) ⊆ 𝑦)
3	2	adantl 275	. . . . 5 ⊢ ((EXMID ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∪ (𝑦 ∖ 𝑥)) ⊆ 𝑦)
4		elun1 3248	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
5	4	adantl 275	. . . . . . . . 9 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
6		simplr 520	. . . . . . . . . . 11 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑦)
7		simpr 109	. . . . . . . . . . 11 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → ¬ 𝑧 ∈ 𝑥)
8	6, 7	eldifd 3086	. . . . . . . . . 10 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝑦 ∖ 𝑥))
9		elun2 3249	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑦 ∖ 𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
10	8, 9	syl 14	. . . . . . . . 9 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
11		exmidexmid 4128	. . . . . . . . . . 11 ⊢ (EXMID → DECID 𝑧 ∈ 𝑥)
12		exmiddc 822	. . . . . . . . . . 11 ⊢ (DECID 𝑧 ∈ 𝑥 → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥))
13	11, 12	syl 14	. . . . . . . . . 10 ⊢ (EXMID → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥))
14	13	adantr 274	. . . . . . . . 9 ⊢ ((EXMID ∧ 𝑧 ∈ 𝑦) → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥))
15	5, 10, 14	mpjaodan 788	. . . . . . . 8 ⊢ ((EXMID ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥)))
16	15	ex 114	. . . . . . 7 ⊢ (EXMID → (𝑧 ∈ 𝑦 → 𝑧 ∈ (𝑥 ∪ (𝑦 ∖ 𝑥))))
17	16	ssrdv 3108	. . . . . 6 ⊢ (EXMID → 𝑦 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥)))
18	17	adantr 274	. . . . 5 ⊢ ((EXMID ∧ 𝑥 ⊆ 𝑦) → 𝑦 ⊆ (𝑥 ∪ (𝑦 ∖ 𝑥)))
19	3, 18	eqssd 3119	. . . 4 ⊢ ((EXMID ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)
20	19	ex 114	. . 3 ⊢ (EXMID → (𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
21	20	alrimivv 1848	. 2 ⊢ (EXMID → ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
22		vex 2692	. . . . . 6 ⊢ 𝑧 ∈ V
23		p0ex 4120	. . . . . 6 ⊢ {∅} ∈ V
24		sseq12 3127	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → (𝑥 ⊆ 𝑦 ↔ 𝑧 ⊆ {∅}))
25		simpl 108	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → 𝑥 = 𝑧)
26		simpr 109	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → 𝑦 = {∅})
27	26, 25	difeq12d 3200	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → (𝑦 ∖ 𝑥) = ({∅} ∖ 𝑧))
28	25, 27	uneq12d 3236	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → (𝑥 ∪ (𝑦 ∖ 𝑥)) = (𝑧 ∪ ({∅} ∖ 𝑧)))
29	28, 26	eqeq12d 2155	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → ((𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦 ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
30	24, 29	imbi12d 233	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = {∅}) → ((𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) ↔ (𝑧 ⊆ {∅} → (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
31	30	spc2gv 2780	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ {∅} ∈ V) → (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
32	22, 23, 31	mp2an 423	. . . . 5 ⊢ (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
33		0ex 4063	. . . . . . . 8 ⊢ ∅ ∈ V
34	33	snid 3563	. . . . . . 7 ⊢ ∅ ∈ {∅}
35		eleq2 2204	. . . . . . 7 ⊢ ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ ∅ ∈ {∅}))
36	34, 35	mpbiri 167	. . . . . 6 ⊢ ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → ∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)))
37		eldifn 3204	. . . . . . . 8 ⊢ (∅ ∈ ({∅} ∖ 𝑧) → ¬ ∅ ∈ 𝑧)
38	37	orim2i 751	. . . . . . 7 ⊢ ((∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)) → (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
39		elun 3222	. . . . . . 7 ⊢ (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ (∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)))
40		df-dc 821	. . . . . . 7 ⊢ (DECID ∅ ∈ 𝑧 ↔ (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
41	38, 39, 40	3imtr4i 200	. . . . . 6 ⊢ (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) → DECID ∅ ∈ 𝑧)
42	36, 41	syl 14	. . . . 5 ⊢ ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → DECID ∅ ∈ 𝑧)
43	32, 42	syl6 33	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
44	43	alrimiv 1847	. . 3 ⊢ (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
45		df-exmid 4127	. . 3 ⊢ (EXMID ↔ ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
46	44, 45	sylibr 133	. 2 ⊢ (∀𝑥∀𝑦(𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) → EXMID)
47	21, 46	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 1330   = wceq 1332   ∈ wcel 1481  Vcvv 2689   ∖ cdif 3073   ∪ cun 3074   ⊆ wss 3076  ∅c0 3368  {csn 3532  EXMIDwem 4126

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-exmid 4127

This theorem is referenced by: (None)