ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  exmidundifim GIF version

Theorem exmidundifim 4100
Description: Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4099 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 3413 . . . . . . 7 (𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
21biimpi 119 . . . . . 6 (𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
32adantl 275 . . . . 5 ((EXMID𝑥𝑦) → (𝑥 ∪ (𝑦𝑥)) ⊆ 𝑦)
4 elun1 3213 . . . . . . . . . 10 (𝑧𝑥𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
54adantl 275 . . . . . . . . 9 (((EXMID𝑧𝑦) ∧ 𝑧𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
6 simplr 504 . . . . . . . . . . 11 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧𝑦)
7 simpr 109 . . . . . . . . . . 11 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → ¬ 𝑧𝑥)
86, 7eldifd 3051 . . . . . . . . . 10 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧 ∈ (𝑦𝑥))
9 elun2 3214 . . . . . . . . . 10 (𝑧 ∈ (𝑦𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
108, 9syl 14 . . . . . . . . 9 (((EXMID𝑧𝑦) ∧ ¬ 𝑧𝑥) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
11 exmidexmid 4090 . . . . . . . . . . 11 (EXMIDDECID 𝑧𝑥)
12 exmiddc 806 . . . . . . . . . . 11 (DECID 𝑧𝑥 → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
1311, 12syl 14 . . . . . . . . . 10 (EXMID → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
1413adantr 274 . . . . . . . . 9 ((EXMID𝑧𝑦) → (𝑧𝑥 ∨ ¬ 𝑧𝑥))
155, 10, 14mpjaodan 772 . . . . . . . 8 ((EXMID𝑧𝑦) → 𝑧 ∈ (𝑥 ∪ (𝑦𝑥)))
1615ex 114 . . . . . . 7 (EXMID → (𝑧𝑦𝑧 ∈ (𝑥 ∪ (𝑦𝑥))))
1716ssrdv 3073 . . . . . 6 (EXMID𝑦 ⊆ (𝑥 ∪ (𝑦𝑥)))
1817adantr 274 . . . . 5 ((EXMID𝑥𝑦) → 𝑦 ⊆ (𝑥 ∪ (𝑦𝑥)))
193, 18eqssd 3084 . . . 4 ((EXMID𝑥𝑦) → (𝑥 ∪ (𝑦𝑥)) = 𝑦)
2019ex 114 . . 3 (EXMID → (𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦))
2120alrimivv 1831 . 2 (EXMID → ∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦))
22 vex 2663 . . . . . 6 𝑧 ∈ V
23 p0ex 4082 . . . . . 6 {∅} ∈ V
24 sseq12 3092 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑥𝑦𝑧 ⊆ {∅}))
25 simpl 108 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = {∅}) → 𝑥 = 𝑧)
26 simpr 109 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = {∅}) → 𝑦 = {∅})
2726, 25difeq12d 3165 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑦𝑥) = ({∅} ∖ 𝑧))
2825, 27uneq12d 3201 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = {∅}) → (𝑥 ∪ (𝑦𝑥)) = (𝑧 ∪ ({∅} ∖ 𝑧)))
2928, 26eqeq12d 2132 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = {∅}) → ((𝑥 ∪ (𝑦𝑥)) = 𝑦 ↔ (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
3024, 29imbi12d 233 . . . . . . 7 ((𝑥 = 𝑧𝑦 = {∅}) → ((𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) ↔ (𝑧 ⊆ {∅} → (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
3130spc2gv 2750 . . . . . 6 ((𝑧 ∈ V ∧ {∅} ∈ V) → (∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅})))
3222, 23, 31mp2an 422 . . . . 5 (∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) → (𝑧 ⊆ {∅} → (𝑧 ∪ ({∅} ∖ 𝑧)) = {∅}))
33 0ex 4025 . . . . . . . 8 ∅ ∈ V
3433snid 3526 . . . . . . 7 ∅ ∈ {∅}
35 eleq2 2181 . . . . . . 7 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ ∅ ∈ {∅}))
3634, 35mpbiri 167 . . . . . 6 ((𝑧 ∪ ({∅} ∖ 𝑧)) = {∅} → ∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)))
37 eldifn 3169 . . . . . . . 8 (∅ ∈ ({∅} ∖ 𝑧) → ¬ ∅ ∈ 𝑧)
3837orim2i 735 . . . . . . 7 ((∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)) → (∅ ∈ 𝑧 ∨ ¬ ∅ ∈ 𝑧))
39 elun 3187 . . . . . . 7 (∅ ∈ (𝑧 ∪ ({∅} ∖ 𝑧)) ↔ (∅ ∈ 𝑧 ∨ ∅ ∈ ({∅} ∖ 𝑧)))
40 df-dc 805 . . . . . . 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 1830 . . 3 (∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦) → ∀𝑧(𝑧 ⊆ {∅} → DECID ∅ ∈ 𝑧))
45 df-exmid 4089 . . 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 682  DECID wdc 804  wal 1314   = wceq 1316  wcel 1465  Vcvv 2660  cdif 3038  cun 3039  wss 3041  c0 3333  {csn 3497  EXMIDwem 4088
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-dc 805  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-exmid 4089
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator