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Theorem exmidundif 4208
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 3505 and undifdcss 6924 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 3505 . . . . . . . 8 (𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
21biimpi 120 . . . . . . 7 (𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
32adantl 277 . . . . . 6 ((EXMID𝑥𝑦) → (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
4 elun1 3304 . . . . . . . . . . 11 (𝑧𝑥𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
54adantl 277 . . . . . . . . . 10 (((EXMID𝑧𝑦) ∧ 𝑧𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
6 simplr 528 . . . . . . . . . . . 12 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧𝑦)
7 simpr 110 . . . . . . . . . . . 12 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → ¬ 𝑧𝑥)
86, 7eldifd 3141 . . . . . . . . . . 11 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧 ∈ (𝑦𝑥))
9 elun2 3305 . . . . . . . . . . 11 (𝑧 ∈ (𝑦𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
108, 9syl 14 . . . . . . . . . 10 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
11 exmidexmid 4198 . . . . . . . . . . . 12 (EXMIDDECID 𝑧𝑥)
12 exmiddc 836 . . . . . . . . . . . 12 (DECID 𝑧𝑥 → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
1311, 12syl 14 . . . . . . . . . . 11 (EXMID → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
1413adantr 276 . . . . . . . . . 10 ((EXMID𝑧𝑦) → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
155, 10, 14mpjaodan 798 . . . . . . . . 9 ((EXMID𝑧𝑦) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
1615ex 115 . . . . . . . 8 (EXMID → (𝑧𝑦𝑧 ∈ (𝑥 ∪ (𝑦𝑥))))
1716ssrdv 3163 . . . . . . 7 (EXMID𝑦 ⊆ (𝑥 ∪ (𝑦𝑥)))
1817adantr 276 . . . . . 6 ((EXMID𝑥𝑦) → 𝑦 ⊆ (𝑥 ∪ (𝑦𝑥)))
193, 18eqssd 3174 . . . . 5 ((EXMID𝑥𝑦) → (𝑥 ∪ (𝑦𝑥)) = 𝑦)
2019ex 115 . . . 4 (EXMID → (𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦))
21 ssun1 3300 . . . . 5 𝑥 ⊆ (𝑥 ∪ (𝑦𝑥))
22 sseq2 3181 . . . . 5 ((𝑥 ∪ (𝑦𝑥)) = 𝑦 → (𝑥 ⊆ (𝑥 ∪ (𝑦𝑥)) ↔ 𝑥𝑦))
2321, 22mpbii 148 . . . 4 ((𝑥 ∪ (𝑦𝑥)) = 𝑦𝑥𝑦)
2420, 23impbid1 142 . . 3 (EXMID → (𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦))
2524alrimivv 1875 . 2 (EXMID → ∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦))
26 vex 2742 . . . . . 6 𝑧 ∈ V
27 p0ex 4190 . . . . . 6 {∅} ∈ V
28 sseq12 3182 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑥𝑦𝑧 ⊆ {∅}))
29 simpl 109 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = {∅}) → 𝑥 = 𝑧)
30 simpr 110 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = {∅}) → 𝑦 = {∅})
3130, 29difeq12d 3256 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑦𝑥) = ({∅} ∖ 𝑧))
3229, 31uneq12d 3292 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑥 ∪ (𝑦𝑥)) = (𝑧 ∪ ({∅} ∖ 𝑧)))
3332, 30eqeq12d 2192 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = {∅}) → ((𝑥 ∪ (𝑦𝑥)) = 𝑦 ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
3428, 33bibi12d 235 . . . . . . 7 ((𝑥 = 𝑧𝑦 = {∅}) → ((𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) ↔ (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
3534spc2gv 2830 . . . . . 6 ((𝑧 ∈ V ∧ {∅} ∈ V) → (∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
3626, 27, 35mp2an 426 . . . . 5 (∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
37 0ex 4132 . . . . . . . 8 ∅ ∈ V
3837snid 3625 . . . . . . 7 ∅ ∈ {∅}
39 eleq2 2241 . . . . . . 7 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ ∅ ∈ {∅}))
4038, 39mpbiri 168 . . . . . 6 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → ∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)))
41 eldifn 3260 . . . . . . . 8 (∅ ∈ ({∅} ∖ 𝑧) → ¬ ∅ ∈ 𝑧)
4241orim2i 761 . . . . . . 7 ((∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)) → (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
43 elun 3278 . . . . . . 7 (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ (∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)))
44 df-dc 835 . . . . . . 7 (DECID ∅ ∈ 𝑧 ↔ (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
4542, 43, 443imtr4i 201 . . . . . 6 (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) → DECID ∅ ∈ 𝑧)
4640, 45syl 14 . . . . 5 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → DECID ∅ ∈ 𝑧)
4736, 46biimtrdi 163 . . . 4 (∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
4847alrimiv 1874 . . 3 (∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) → ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
49 df-exmid 4197 . . 3 (EXMID ↔ ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
5048, 49sylibr 134 . 2 (∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) → EXMID)
5125, 50impbii 126 1 (EXMID ↔ ∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  wal 1351   = wceq 1353  wcel 2148  Vcvv 2739  cdif 3128  cun 3129  wss 3131  c0 3424  {csn 3594  EXMIDwem 4196
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-exmid 4197
This theorem is referenced by: (None)
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