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Theorem exmidundifim 4191
Description: Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4190 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.)
Assertion
Ref Expression
exmidundifim (EXMID ↔ ∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦))
Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidundifim
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 undifss 3494 . . . . . . 7 (𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
21biimpi 119 . . . . . 6 (𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
32adantl 275 . . . . 5 ((EXMID𝑥𝑦) → (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
4 elun1 3294 . . . . . . . . . 10 (𝑧𝑥𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
54adantl 275 . . . . . . . . 9 (((EXMID𝑧𝑦) ∧ 𝑧𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
6 simplr 525 . . . . . . . . . . 11 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧𝑦)
7 simpr 109 . . . . . . . . . . 11 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → ¬ 𝑧𝑥)
86, 7eldifd 3131 . . . . . . . . . 10 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧 ∈ (𝑦𝑥))
9 elun2 3295 . . . . . . . . . 10 (𝑧 ∈ (𝑦𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
108, 9syl 14 . . . . . . . . 9 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
11 exmidexmid 4180 . . . . . . . . . . 11 (EXMIDDECID 𝑧𝑥)
12 exmiddc 831 . . . . . . . . . . 11 (DECID 𝑧𝑥 → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
1311, 12syl 14 . . . . . . . . . 10 (EXMID → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
1413adantr 274 . . . . . . . . 9 ((EXMID𝑧𝑦) → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
155, 10, 14mpjaodan 793 . . . . . . . 8 ((EXMID𝑧𝑦) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
1615ex 114 . . . . . . 7 (EXMID → (𝑧𝑦𝑧 ∈ (𝑥 ∪ (𝑦𝑥))))
1716ssrdv 3153 . . . . . 6 (EXMID𝑦 ⊆ (𝑥 ∪ (𝑦𝑥)))
1817adantr 274 . . . . 5 ((EXMID𝑥𝑦) → 𝑦 ⊆ (𝑥 ∪ (𝑦𝑥)))
193, 18eqssd 3164 . . . 4 ((EXMID𝑥𝑦) → (𝑥 ∪ (𝑦𝑥)) = 𝑦)
2019ex 114 . . 3 (EXMID → (𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦))
2120alrimivv 1868 . 2 (EXMID → ∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦))
22 vex 2733 . . . . . 6 𝑧 ∈ V
23 p0ex 4172 . . . . . 6 {∅} ∈ V
24 sseq12 3172 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑥𝑦𝑧 ⊆ {∅}))
25 simpl 108 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = {∅}) → 𝑥 = 𝑧)
26 simpr 109 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = {∅}) → 𝑦 = {∅})
2726, 25difeq12d 3246 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑦𝑥) = ({∅} ∖ 𝑧))
2825, 27uneq12d 3282 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑥 ∪ (𝑦𝑥)) = (𝑧 ∪ ({∅} ∖ 𝑧)))
2928, 26eqeq12d 2185 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = {∅}) → ((𝑥 ∪ (𝑦𝑥)) = 𝑦 ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
3024, 29imbi12d 233 . . . . . . 7 ((𝑥 = 𝑧𝑦 = {∅}) → ((𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) ↔ (𝑧 ⊆ {∅} → (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
3130spc2gv 2821 . . . . . 6 ((𝑧 ∈ V ∧ {∅} ∈ V) → (∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
3222, 23, 31mp2an 424 . . . . 5 (∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
33 0ex 4114 . . . . . . . 8 ∅ ∈ V
3433snid 3612 . . . . . . 7 ∅ ∈ {∅}
35 eleq2 2234 . . . . . . 7 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ ∅ ∈ {∅}))
3634, 35mpbiri 167 . . . . . 6 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → ∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)))
37 eldifn 3250 . . . . . . . 8 (∅ ∈ ({∅} ∖ 𝑧) → ¬ ∅ ∈ 𝑧)
3837orim2i 756 . . . . . . 7 ((∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)) → (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
39 elun 3268 . . . . . . 7 (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ (∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)))
40 df-dc 830 . . . . . . 7 (DECID ∅ ∈ 𝑧 ↔ (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
4138, 39, 403imtr4i 200 . . . . . 6 (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) → DECID ∅ ∈ 𝑧)
4236, 41syl 14 . . . . 5 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → DECID ∅ ∈ 𝑧)
4332, 42syl6 33 . . . 4 (∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
4443alrimiv 1867 . . 3 (∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) → ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
45 df-exmid 4179 . . 3 (EXMID ↔ ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
4644, 45sylibr 133 . 2 (∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) → EXMID)
4721, 46impbii 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 703  DECID wdc 829  wal 1346   = wceq 1348  wcel 2141  Vcvv 2730  cdif 3118  cun 3119  wss 3121  c0 3414  {csn 3581  EXMIDwem 4178
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-exmid 4179
This theorem is referenced by: (None)
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