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Theorem exmid01 4116
 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 4114 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
2 df-dc 820 . . . . 5 (DECID ∅ ∈ 𝑥 ↔ (∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥))
3 orcom 717 . . . . . 6 ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥))
4 simpll 518 . . . . . . . . . . . . . 14 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑥 ⊆ {∅})
5 simpr 109 . . . . . . . . . . . . . 14 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦𝑥)
64, 5sseldd 3093 . . . . . . . . . . . . 13 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦 ∈ {∅})
7 velsn 3539 . . . . . . . . . . . . 13 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
86, 7sylib 121 . . . . . . . . . . . 12 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦 = ∅)
98, 5eqeltrrd 2215 . . . . . . . . . . 11 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → ∅ ∈ 𝑥)
10 simplr 519 . . . . . . . . . . 11 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → ¬ ∅ ∈ 𝑥)
119, 10pm2.65da 650 . . . . . . . . . 10 ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → ¬ 𝑦𝑥)
1211eq0rdv 3402 . . . . . . . . 9 ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → 𝑥 = ∅)
1312ex 114 . . . . . . . 8 (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥𝑥 = ∅))
14 noel 3362 . . . . . . . . 9 ¬ ∅ ∈ ∅
15 eleq2 2201 . . . . . . . . 9 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
1614, 15mtbiri 664 . . . . . . . 8 (𝑥 = ∅ → ¬ ∅ ∈ 𝑥)
1713, 16impbid1 141 . . . . . . 7 (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥𝑥 = ∅))
18 elex2 2697 . . . . . . . . . 10 (∅ ∈ 𝑥 → ∃𝑧 𝑧𝑥)
19 sssnm 3676 . . . . . . . . . 10 (∃𝑧 𝑧𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
2018, 19syl 14 . . . . . . . . 9 (∅ ∈ 𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
2120biimpcd 158 . . . . . . . 8 (𝑥 ⊆ {∅} → (∅ ∈ 𝑥𝑥 = {∅}))
22 0ex 4050 . . . . . . . . . 10 ∅ ∈ V
2322snid 3551 . . . . . . . . 9 ∅ ∈ {∅}
24 eleq2 2201 . . . . . . . . 9 (𝑥 = {∅} → (∅ ∈ 𝑥 ↔ ∅ ∈ {∅}))
2523, 24mpbiri 167 . . . . . . . 8 (𝑥 = {∅} → ∅ ∈ 𝑥)
2621, 25impbid1 141 . . . . . . 7 (𝑥 ⊆ {∅} → (∅ ∈ 𝑥𝑥 = {∅}))
2717, 26orbi12d 782 . . . . . 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 1446 . 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 697  DECID wdc 819  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480   ⊆ wss 3066  ∅c0 3358  {csn 3522  EXMIDwem 4113 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-nul 4049 This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359  df-sn 3528  df-exmid 4114 This theorem is referenced by:  exmid1dc  4118  exmidn0m  4119  exmidsssn  4120  exmidpw  6795  exmidomni  7007  ss1oel2o  13178  exmidsbthrlem  13206  sbthom  13210
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