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Theorem exmidundif 4129
 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 3443 and undifdcss 6811 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 ↔ ∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦))
Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidundif
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 undifss 3443 . . . . . . . 8 (𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
21biimpi 119 . . . . . . 7 (𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
32adantl 275 . . . . . 6 ((EXMID𝑥𝑦) → (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
4 elun1 3243 . . . . . . . . . . 11 (𝑧𝑥𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
54adantl 275 . . . . . . . . . 10 (((EXMID𝑧𝑦) ∧ 𝑧𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
6 simplr 519 . . . . . . . . . . . 12 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧𝑦)
7 simpr 109 . . . . . . . . . . . 12 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → ¬ 𝑧𝑥)
86, 7eldifd 3081 . . . . . . . . . . 11 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧 ∈ (𝑦𝑥))
9 elun2 3244 . . . . . . . . . . 11 (𝑧 ∈ (𝑦𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
108, 9syl 14 . . . . . . . . . 10 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
11 exmidexmid 4120 . . . . . . . . . . . 12 (EXMIDDECID 𝑧𝑥)
12 exmiddc 821 . . . . . . . . . . . 12 (DECID 𝑧𝑥 → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
1311, 12syl 14 . . . . . . . . . . 11 (EXMID → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
1413adantr 274 . . . . . . . . . 10 ((EXMID𝑧𝑦) → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
155, 10, 14mpjaodan 787 . . . . . . . . 9 ((EXMID𝑧𝑦) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
1615ex 114 . . . . . . . 8 (EXMID → (𝑧𝑦𝑧 ∈ (𝑥 ∪ (𝑦𝑥))))
1716ssrdv 3103 . . . . . . 7 (EXMID𝑦 ⊆ (𝑥 ∪ (𝑦𝑥)))
1817adantr 274 . . . . . 6 ((EXMID𝑥𝑦) → 𝑦 ⊆ (𝑥 ∪ (𝑦𝑥)))
193, 18eqssd 3114 . . . . 5 ((EXMID𝑥𝑦) → (𝑥 ∪ (𝑦𝑥)) = 𝑦)
2019ex 114 . . . 4 (EXMID → (𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦))
21 ssun1 3239 . . . . 5 𝑥 ⊆ (𝑥 ∪ (𝑦𝑥))
22 sseq2 3121 . . . . 5 ((𝑥 ∪ (𝑦𝑥)) = 𝑦 → (𝑥 ⊆ (𝑥 ∪ (𝑦𝑥)) ↔ 𝑥𝑦))
2321, 22mpbii 147 . . . 4 ((𝑥 ∪ (𝑦𝑥)) = 𝑦𝑥𝑦)
2420, 23impbid1 141 . . 3 (EXMID → (𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦))
2524alrimivv 1847 . 2 (EXMID → ∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦))
26 vex 2689 . . . . . 6 𝑧 ∈ V
27 p0ex 4112 . . . . . 6 {∅} ∈ V
28 sseq12 3122 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑥𝑦𝑧 ⊆ {∅}))
29 simpl 108 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = {∅}) → 𝑥 = 𝑧)
30 simpr 109 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = {∅}) → 𝑦 = {∅})
3130, 29difeq12d 3195 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑦𝑥) = ({∅} ∖ 𝑧))
3229, 31uneq12d 3231 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑥 ∪ (𝑦𝑥)) = (𝑧 ∪ ({∅} ∖ 𝑧)))
3332, 30eqeq12d 2154 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = {∅}) → ((𝑥 ∪ (𝑦𝑥)) = 𝑦 ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
3428, 33bibi12d 234 . . . . . . 7 ((𝑥 = 𝑧𝑦 = {∅}) → ((𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) ↔ (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
3534spc2gv 2776 . . . . . 6 ((𝑧 ∈ V ∧ {∅} ∈ V) → (∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
3626, 27, 35mp2an 422 . . . . 5 (∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
37 0ex 4055 . . . . . . . 8 ∅ ∈ V
3837snid 3556 . . . . . . 7 ∅ ∈ {∅}
39 eleq2 2203 . . . . . . 7 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ ∅ ∈ {∅}))
4038, 39mpbiri 167 . . . . . 6 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → ∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)))
41 eldifn 3199 . . . . . . . 8 (∅ ∈ ({∅} ∖ 𝑧) → ¬ ∅ ∈ 𝑧)
4241orim2i 750 . . . . . . 7 ((∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)) → (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
43 elun 3217 . . . . . . 7 (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ (∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)))
44 df-dc 820 . . . . . . 7 (DECID ∅ ∈ 𝑧 ↔ (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
4542, 43, 443imtr4i 200 . . . . . 6 (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) → DECID ∅ ∈ 𝑧)
4640, 45syl 14 . . . . 5 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → DECID ∅ ∈ 𝑧)
4736, 46syl6bi 162 . . . 4 (∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
4847alrimiv 1846 . . 3 (∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) → ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
49 df-exmid 4119 . . 3 (EXMID ↔ ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
5048, 49sylibr 133 . 2 (∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) → EXMID)
5125, 50impbii 125 1 (EXMID ↔ ∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  DECID wdc 819  ∀wal 1329   = wceq 1331   ∈ wcel 1480  Vcvv 2686   ∖ cdif 3068   ∪ cun 3069   ⊆ wss 3071  ∅c0 3363  {csn 3527  EXMIDwem 4118 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098 This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-exmid 4119 This theorem is referenced by: (None)
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