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Theorem exmid1dc 4091
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 4085 or ordtriexmid 4405. 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 804 . . . . . . 7 (DECID 𝑥 = {∅} → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
31, 2syl 14 . . . . . 6 ((𝜑𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
4 df-ne 2284 . . . . . . . . 9 (𝑥 ≠ {∅} ↔ ¬ 𝑥 = {∅})
5 pwntru 4090 . . . . . . . . . 10 ((𝑥 ⊆ {∅} ∧ 𝑥 ≠ {∅}) → 𝑥 = ∅)
65ex 114 . . . . . . . . 9 (𝑥 ⊆ {∅} → (𝑥 ≠ {∅} → 𝑥 = ∅))
74, 6syl5bir 152 . . . . . . . 8 (𝑥 ⊆ {∅} → (¬ 𝑥 = {∅} → 𝑥 = ∅))
87orim2d 760 . . . . . . 7 (𝑥 ⊆ {∅} → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
98adantl 273 . . . . . 6 ((𝜑𝑥 ⊆ {∅}) → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
103, 9mpd 13 . . . . 5 ((𝜑𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅))
1110orcomd 701 . . . 4 ((𝜑𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
1211ex 114 . . 3 (𝜑 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1312alrimiv 1828 . 2 (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14 exmid01 4089 . 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 680  DECID wdc 802  wal 1312   = wceq 1314  wne 2283  wss 3039  c0 3331  {csn 3495  EXMIDwem 4086
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022
This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-exmid 4087
This theorem is referenced by:  exmidaclem  7028
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