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Theorem exmid1dc 4161
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 4154 or ordtriexmid 4480. 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 822 . . . . . . 7 (DECID 𝑥 = {∅} → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
31, 2syl 14 . . . . . 6 ((𝜑𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
4 df-ne 2328 . . . . . . . . 9 (𝑥 ≠ {∅} ↔ ¬ 𝑥 = {∅})
5 pwntru 4160 . . . . . . . . . 10 ((𝑥 ⊆ {∅} ∧ 𝑥 ≠ {∅}) → 𝑥 = ∅)
65ex 114 . . . . . . . . 9 (𝑥 ⊆ {∅} → (𝑥 ≠ {∅} → 𝑥 = ∅))
74, 6syl5bir 152 . . . . . . . 8 (𝑥 ⊆ {∅} → (¬ 𝑥 = {∅} → 𝑥 = ∅))
87orim2d 778 . . . . . . 7 (𝑥 ⊆ {∅} → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
98adantl 275 . . . . . 6 ((𝜑𝑥 ⊆ {∅}) → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
103, 9mpd 13 . . . . 5 ((𝜑𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅))
1110orcomd 719 . . . 4 ((𝜑𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
1211ex 114 . . 3 (𝜑 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1312alrimiv 1854 . 2 (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14 exmid01 4159 . 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 698  DECID wdc 820  wal 1333   = wceq 1335  wne 2327  wss 3102  c0 3394  {csn 3560  EXMIDwem 4155
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-nul 4090
This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566  df-exmid 4156
This theorem is referenced by:  pw1fin  6855  exmidonfin  7129  exmidaclem  7143  exmidontri  7174  exmidontri2or  7178
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