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Theorem exmid01 4053
Description: Excluded middle is equivalent to saying any subset of {∅} is either or {∅}. (Contributed by BJ and Jim Kingdon, 18-Jun-2022.)
Assertion
Ref Expression
exmid01 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))

Proof of Theorem exmid01
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-exmid 4051 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
2 df-dc 784 . . . . 5 (DECID ∅ ∈ 𝑥 ↔ (∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥))
3 orcom 685 . . . . . 6 ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥))
4 simpll 497 . . . . . . . . . . . . . 14 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑥 ⊆ {∅})
5 simpr 109 . . . . . . . . . . . . . 14 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦𝑥)
64, 5sseldd 3040 . . . . . . . . . . . . 13 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦 ∈ {∅})
7 velsn 3483 . . . . . . . . . . . . 13 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
86, 7sylib 121 . . . . . . . . . . . 12 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦 = ∅)
98, 5eqeltrrd 2172 . . . . . . . . . . 11 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → ∅ ∈ 𝑥)
10 simplr 498 . . . . . . . . . . 11 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → ¬ ∅ ∈ 𝑥)
119, 10pm2.65da 625 . . . . . . . . . 10 ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → ¬ 𝑦𝑥)
1211eq0rdv 3346 . . . . . . . . 9 ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → 𝑥 = ∅)
1312ex 114 . . . . . . . 8 (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥𝑥 = ∅))
14 noel 3306 . . . . . . . . 9 ¬ ∅ ∈ ∅
15 eleq2 2158 . . . . . . . . 9 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
1614, 15mtbiri 638 . . . . . . . 8 (𝑥 = ∅ → ¬ ∅ ∈ 𝑥)
1713, 16impbid1 141 . . . . . . 7 (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥𝑥 = ∅))
18 elex2 2649 . . . . . . . . . 10 (∅ ∈ 𝑥 → ∃𝑧 𝑧𝑥)
19 sssnm 3620 . . . . . . . . . 10 (∃𝑧 𝑧𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
2018, 19syl 14 . . . . . . . . 9 (∅ ∈ 𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
2120biimpcd 158 . . . . . . . 8 (𝑥 ⊆ {∅} → (∅ ∈ 𝑥𝑥 = {∅}))
22 0ex 3987 . . . . . . . . . 10 ∅ ∈ V
2322snid 3495 . . . . . . . . 9 ∅ ∈ {∅}
24 eleq2 2158 . . . . . . . . 9 (𝑥 = {∅} → (∅ ∈ 𝑥 ↔ ∅ ∈ {∅}))
2523, 24mpbiri 167 . . . . . . . 8 (𝑥 = {∅} → ∅ ∈ 𝑥)
2621, 25impbid1 141 . . . . . . 7 (𝑥 ⊆ {∅} → (∅ ∈ 𝑥𝑥 = {∅}))
2717, 26orbi12d 745 . . . . . 6 (𝑥 ⊆ {∅} → ((¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
283, 27syl5bb 191 . . . . 5 (𝑥 ⊆ {∅} → ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
292, 28syl5bb 191 . . . 4 (𝑥 ⊆ {∅} → (DECID ∅ ∈ 𝑥 ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
3029pm5.74i 179 . . 3 ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
3130albii 1411 . 2 (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
321, 31bitri 183 1 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 667  DECID wdc 783  wal 1294   = wceq 1296  wex 1433  wcel 1445  wss 3013  c0 3302  {csn 3466  EXMIDwem 4050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-nul 3986
This theorem depends on definitions:  df-bi 116  df-dc 784  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015  df-in 3019  df-ss 3026  df-nul 3303  df-sn 3472  df-exmid 4051
This theorem is referenced by:  exmidn0m  4054  exmidsssn  4055  exmidpw  6704  exmidomni  6885  ss1oel2o  12610  exmidsbthrlem  12633
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