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Theorem exmidundifim 4219
Description: Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4218 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 3515 . . . . . . 7 (𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
21biimpi 120 . . . . . 6 (𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
32adantl 277 . . . . 5 ((EXMID𝑥𝑦) → (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
4 elun1 3314 . . . . . . . . . 10 (𝑧𝑥𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
54adantl 277 . . . . . . . . 9 (((EXMID𝑧𝑦) ∧ 𝑧𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
6 simplr 528 . . . . . . . . . . 11 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧𝑦)
7 simpr 110 . . . . . . . . . . 11 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → ¬ 𝑧𝑥)
86, 7eldifd 3151 . . . . . . . . . 10 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧 ∈ (𝑦𝑥))
9 elun2 3315 . . . . . . . . . 10 (𝑧 ∈ (𝑦𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
108, 9syl 14 . . . . . . . . 9 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
11 exmidexmid 4208 . . . . . . . . . . 11 (EXMIDDECID 𝑧𝑥)
12 exmiddc 837 . . . . . . . . . . 11 (DECID 𝑧𝑥 → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
1311, 12syl 14 . . . . . . . . . 10 (EXMID → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
1413adantr 276 . . . . . . . . 9 ((EXMID𝑧𝑦) → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
155, 10, 14mpjaodan 799 . . . . . . . 8 ((EXMID𝑧𝑦) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
1615ex 115 . . . . . . 7 (EXMID → (𝑧𝑦𝑧 ∈ (𝑥 ∪ (𝑦𝑥))))
1716ssrdv 3173 . . . . . 6 (EXMID𝑦 ⊆ (𝑥 ∪ (𝑦𝑥)))
1817adantr 276 . . . . 5 ((EXMID𝑥𝑦) → 𝑦 ⊆ (𝑥 ∪ (𝑦𝑥)))
193, 18eqssd 3184 . . . 4 ((EXMID𝑥𝑦) → (𝑥 ∪ (𝑦𝑥)) = 𝑦)
2019ex 115 . . 3 (EXMID → (𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦))
2120alrimivv 1885 . 2 (EXMID → ∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦))
22 vex 2752 . . . . . 6 𝑧 ∈ V
23 p0ex 4200 . . . . . 6 {∅} ∈ V
24 sseq12 3192 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑥𝑦𝑧 ⊆ {∅}))
25 simpl 109 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = {∅}) → 𝑥 = 𝑧)
26 simpr 110 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = {∅}) → 𝑦 = {∅})
2726, 25difeq12d 3266 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑦𝑥) = ({∅} ∖ 𝑧))
2825, 27uneq12d 3302 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑥 ∪ (𝑦𝑥)) = (𝑧 ∪ ({∅} ∖ 𝑧)))
2928, 26eqeq12d 2202 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = {∅}) → ((𝑥 ∪ (𝑦𝑥)) = 𝑦 ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
3024, 29imbi12d 234 . . . . . . 7 ((𝑥 = 𝑧𝑦 = {∅}) → ((𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) ↔ (𝑧 ⊆ {∅} → (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
3130spc2gv 2840 . . . . . 6 ((𝑧 ∈ V ∧ {∅} ∈ V) → (∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
3222, 23, 31mp2an 426 . . . . 5 (∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
33 0ex 4142 . . . . . . . 8 ∅ ∈ V
3433snid 3635 . . . . . . 7 ∅ ∈ {∅}
35 eleq2 2251 . . . . . . 7 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ ∅ ∈ {∅}))
3634, 35mpbiri 168 . . . . . 6 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → ∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)))
37 eldifn 3270 . . . . . . . 8 (∅ ∈ ({∅} ∖ 𝑧) → ¬ ∅ ∈ 𝑧)
3837orim2i 762 . . . . . . 7 ((∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)) → (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
39 elun 3288 . . . . . . 7 (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ (∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)))
40 df-dc 836 . . . . . . 7 (DECID ∅ ∈ 𝑧 ↔ (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
4138, 39, 403imtr4i 201 . . . . . 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 1884 . . 3 (∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) → ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
45 df-exmid 4207 . . 3 (EXMID ↔ ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
4644, 45sylibr 134 . 2 (∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) → EXMID)
4721, 46impbii 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 709  DECID wdc 835  wal 1361   = wceq 1363  wcel 2158  Vcvv 2749  cdif 3138  cun 3139  wss 3141  c0 3434  {csn 3604  EXMIDwem 4206
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-exmid 4207
This theorem is referenced by: (None)
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