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Theorem exmidpweq 6854
Description: Excluded middle is equivalent to the power set of 1o being 2o. (Contributed by Jim Kingdon, 28-Jul-2024.)
Assertion
Ref Expression
exmidpweq (EXMID ↔ 𝒫 1o = 2o)

Proof of Theorem exmidpweq
StepHypRef Expression
1 exmid01 4159 . . . . . . . 8 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
21biimpi 119 . . . . . . 7 (EXMID → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
3219.21bi 1538 . . . . . 6 (EXMID → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
4 df1o2 6376 . . . . . . . . 9 1o = {∅}
54pweqi 3547 . . . . . . . 8 𝒫 1o = 𝒫 {∅}
65eleq2i 2224 . . . . . . 7 (𝑥 ∈ 𝒫 1o𝑥 ∈ 𝒫 {∅})
7 velpw 3550 . . . . . . 7 (𝑥 ∈ 𝒫 {∅} ↔ 𝑥 ⊆ {∅})
86, 7bitri 183 . . . . . 6 (𝑥 ∈ 𝒫 1o𝑥 ⊆ {∅})
9 vex 2715 . . . . . . 7 𝑥 ∈ V
109elpr 3581 . . . . . 6 (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
113, 8, 103imtr4g 204 . . . . 5 (EXMID → (𝑥 ∈ 𝒫 1o𝑥 ∈ {∅, {∅}}))
1211ssrdv 3134 . . . 4 (EXMID → 𝒫 1o ⊆ {∅, {∅}})
13 pwpw0ss 3767 . . . . . 6 {∅, {∅}} ⊆ 𝒫 {∅}
1413, 5sseqtrri 3163 . . . . 5 {∅, {∅}} ⊆ 𝒫 1o
1514a1i 9 . . . 4 (EXMID → {∅, {∅}} ⊆ 𝒫 1o)
1612, 15eqssd 3145 . . 3 (EXMID → 𝒫 1o = {∅, {∅}})
17 df2o2 6378 . . 3 2o = {∅, {∅}}
1816, 17eqtr4di 2208 . 2 (EXMID → 𝒫 1o = 2o)
19 simpr 109 . . . . . . . . 9 ((𝒫 1o = 2o𝑥 ⊆ {∅}) → 𝑥 ⊆ {∅})
2019, 7sylibr 133 . . . . . . . 8 ((𝒫 1o = 2o𝑥 ⊆ {∅}) → 𝑥 ∈ 𝒫 {∅})
2120, 5eleqtrrdi 2251 . . . . . . 7 ((𝒫 1o = 2o𝑥 ⊆ {∅}) → 𝑥 ∈ 𝒫 1o)
22 simpl 108 . . . . . . . 8 ((𝒫 1o = 2o𝑥 ⊆ {∅}) → 𝒫 1o = 2o)
2322, 17eqtrdi 2206 . . . . . . 7 ((𝒫 1o = 2o𝑥 ⊆ {∅}) → 𝒫 1o = {∅, {∅}})
2421, 23eleqtrd 2236 . . . . . 6 ((𝒫 1o = 2o𝑥 ⊆ {∅}) → 𝑥 ∈ {∅, {∅}})
2524, 10sylib 121 . . . . 5 ((𝒫 1o = 2o𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
2625ex 114 . . . 4 (𝒫 1o = 2o → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2726alrimiv 1854 . . 3 (𝒫 1o = 2o → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2827, 1sylibr 133 . 2 (𝒫 1o = 2oEXMID)
2918, 28impbii 125 1 (EXMID ↔ 𝒫 1o = 2o)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 698  wal 1333   = wceq 1335  wcel 2128  wss 3102  c0 3394  𝒫 cpw 3543  {csn 3560  {cpr 3561  EXMIDwem 4155  1oc1o 6356  2oc2o 6357
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-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-exmid 4156  df-suc 4331  df-1o 6363  df-2o 6364
This theorem is referenced by:  pw1fin  6855  pw1nel3  7166  3nsssucpw1  7171  onntri35  7172
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