ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  exmid1dc GIF version

Theorem exmid1dc 4186
Description: A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4179 or ordtriexmid 4505. In this context 𝑥 = {∅} can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
Hypothesis
Ref Expression
exmid1dc.x ((𝜑𝑥 ⊆ {∅}) → DECID 𝑥 = {∅})
Assertion
Ref Expression
exmid1dc (𝜑EXMID)
Distinct variable group:   𝜑,𝑥

Proof of Theorem exmid1dc
StepHypRef Expression
1 exmid1dc.x . . . . . . 7 ((𝜑𝑥 ⊆ {∅}) → DECID 𝑥 = {∅})
2 exmiddc 831 . . . . . . 7 (DECID 𝑥 = {∅} → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
31, 2syl 14 . . . . . 6 ((𝜑𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
4 df-ne 2341 . . . . . . . . 9 (𝑥 ≠ {∅} ↔ ¬ 𝑥 = {∅})
5 pwntru 4185 . . . . . . . . . 10 ((𝑥 ⊆ {∅} ∧ 𝑥 ≠ {∅}) → 𝑥 = ∅)
65ex 114 . . . . . . . . 9 (𝑥 ⊆ {∅} → (𝑥 ≠ {∅} → 𝑥 = ∅))
74, 6syl5bir 152 . . . . . . . 8 (𝑥 ⊆ {∅} → (¬ 𝑥 = {∅} → 𝑥 = ∅))
87orim2d 783 . . . . . . 7 (𝑥 ⊆ {∅} → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
98adantl 275 . . . . . 6 ((𝜑𝑥 ⊆ {∅}) → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
103, 9mpd 13 . . . . 5 ((𝜑𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅))
1110orcomd 724 . . . 4 ((𝜑𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
1211ex 114 . . 3 (𝜑 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1312alrimiv 1867 . 2 (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14 exmid01 4184 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1513, 14sylibr 133 1 (𝜑EXMID)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 703  DECID wdc 829  wal 1346   = wceq 1348  wne 2340  wss 3121  c0 3414  {csn 3583  EXMIDwem 4180
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-ext 2152  ax-nul 4115
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-exmid 4181
This theorem is referenced by:  pw1fin  6888  exmidonfin  7171  exmidaclem  7185  exmidontri  7216  exmidontri2or  7220
  Copyright terms: Public domain W3C validator