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Theorem exmidundif 4203
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 3503 and undifdcss 6916 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 3503 . . . . . . . 8 (𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
21biimpi 120 . . . . . . 7 (𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
32adantl 277 . . . . . 6 ((EXMID𝑥𝑦) → (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
4 elun1 3302 . . . . . . . . . . 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 3139 . . . . . . . . . . 11 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧 ∈ (𝑦𝑥))
9 elun2 3303 . . . . . . . . . . 11 (𝑧 ∈ (𝑦𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
108, 9syl 14 . . . . . . . . . 10 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
11 exmidexmid 4193 . . . . . . . . . . . 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 3161 . . . . . . 7 (EXMID𝑦 ⊆ (𝑥 ∪ (𝑦𝑥)))
1817adantr 276 . . . . . 6 ((EXMID𝑥𝑦) → 𝑦 ⊆ (𝑥 ∪ (𝑦𝑥)))
193, 18eqssd 3172 . . . . 5 ((EXMID𝑥𝑦) → (𝑥 ∪ (𝑦𝑥)) = 𝑦)
2019ex 115 . . . 4 (EXMID → (𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦))
21 ssun1 3298 . . . . 5 𝑥 ⊆ (𝑥 ∪ (𝑦𝑥))
22 sseq2 3179 . . . . 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 2740 . . . . . 6 𝑧 ∈ V
27 p0ex 4185 . . . . . 6 {∅} ∈ V
28 sseq12 3180 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑥𝑦𝑧 ⊆ {∅}))
29 simpl 109 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = {∅}) → 𝑥 = 𝑧)
30 simpr 110 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = {∅}) → 𝑦 = {∅})
3130, 29difeq12d 3254 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑦𝑥) = ({∅} ∖ 𝑧))
3229, 31uneq12d 3290 . . . . . . . . 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 2828 . . . . . 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 4127 . . . . . . . 8 ∅ ∈ V
3837snid 3622 . . . . . . 7 ∅ ∈ {∅}
39 eleq2 2241 . . . . . . 7 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ ∅ ∈ {∅}))
4038, 39mpbiri 168 . . . . . 6 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → ∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)))
41 eldifn 3258 . . . . . . . 8 (∅ ∈ ({∅} ∖ 𝑧) → ¬ ∅ ∈ 𝑧)
4241orim2i 761 . . . . . . 7 ((∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)) → (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
43 elun 3276 . . . . . . 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, 46syl6bi 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 4192 . . 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 2737  cdif 3126  cun 3127  wss 3129  c0 3422  {csn 3591  EXMIDwem 4191
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 4118  ax-nul 4126  ax-pow 4171
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 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-exmid 4192
This theorem is referenced by: (None)
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