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Theorem exmid1dc 4199
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 4192 or ordtriexmid 4519. 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 836 . . . . . . 7 (DECID 𝑥 = {∅} → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
31, 2syl 14 . . . . . 6 ((𝜑𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
4 df-ne 2348 . . . . . . . . 9 (𝑥 ≠ {∅} ↔ ¬ 𝑥 = {∅})
5 pwntru 4198 . . . . . . . . . 10 ((𝑥 ⊆ {∅} ∧ 𝑥 ≠ {∅}) → 𝑥 = ∅)
65ex 115 . . . . . . . . 9 (𝑥 ⊆ {∅} → (𝑥 ≠ {∅} → 𝑥 = ∅))
74, 6biimtrrid 153 . . . . . . . 8 (𝑥 ⊆ {∅} → (¬ 𝑥 = {∅} → 𝑥 = ∅))
87orim2d 788 . . . . . . 7 (𝑥 ⊆ {∅} → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
98adantl 277 . . . . . 6 ((𝜑𝑥 ⊆ {∅}) → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
103, 9mpd 13 . . . . 5 ((𝜑𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅))
1110orcomd 729 . . . 4 ((𝜑𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
1211ex 115 . . 3 (𝜑 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1312alrimiv 1874 . 2 (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14 exmid01 4197 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1513, 14sylibr 134 1 (𝜑EXMID)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708  DECID wdc 834  wal 1351   = wceq 1353  wne 2347  wss 3129  c0 3422  {csn 3592  EXMIDwem 4193
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4128
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598  df-exmid 4194
This theorem is referenced by:  pw1fin  6907  exmidonfin  7190  exmidaclem  7204  exmidontri  7235  exmidontri2or  7239
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