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Theorem exmidpw 6802
Description: Excluded middle is equivalent to the power set of 1o having two elements. Remark of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 30-Jun-2022.)
Assertion
Ref Expression
exmidpw (EXMID ↔ 𝒫 1o ≈ 2o)

Proof of Theorem exmidpw
StepHypRef Expression
1 df1o2 6326 . . . . 5 1o = {∅}
2 p0ex 4112 . . . . 5 {∅} ∈ V
31, 2eqeltri 2212 . . . 4 1o ∈ V
43pwex 4107 . . 3 𝒫 1o ∈ V
5 exmid01 4121 . . . . . . . . 9 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
65biimpi 119 . . . . . . . 8 (EXMID → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
7619.21bi 1537 . . . . . . 7 (EXMID → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
81pweqi 3514 . . . . . . . . 9 𝒫 1o = 𝒫 {∅}
98eleq2i 2206 . . . . . . . 8 (𝑥 ∈ 𝒫 1o𝑥 ∈ 𝒫 {∅})
10 velpw 3517 . . . . . . . 8 (𝑥 ∈ 𝒫 {∅} ↔ 𝑥 ⊆ {∅})
119, 10bitri 183 . . . . . . 7 (𝑥 ∈ 𝒫 1o𝑥 ⊆ {∅})
12 vex 2689 . . . . . . . 8 𝑥 ∈ V
1312elpr 3548 . . . . . . 7 (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
147, 11, 133imtr4g 204 . . . . . 6 (EXMID → (𝑥 ∈ 𝒫 1o𝑥 ∈ {∅, {∅}}))
1514ssrdv 3103 . . . . 5 (EXMID → 𝒫 1o ⊆ {∅, {∅}})
16 pwpw0ss 3731 . . . . . . 7 {∅, {∅}} ⊆ 𝒫 {∅}
1716, 8sseqtrri 3132 . . . . . 6 {∅, {∅}} ⊆ 𝒫 1o
1817a1i 9 . . . . 5 (EXMID → {∅, {∅}} ⊆ 𝒫 1o)
1915, 18eqssd 3114 . . . 4 (EXMID → 𝒫 1o = {∅, {∅}})
20 df2o2 6328 . . . 4 2o = {∅, {∅}}
2119, 20eqtr4di 2190 . . 3 (EXMID → 𝒫 1o = 2o)
22 eqeng 6660 . . 3 (𝒫 1o ∈ V → (𝒫 1o = 2o → 𝒫 1o ≈ 2o))
234, 21, 22mpsyl 65 . 2 (EXMID → 𝒫 1o ≈ 2o)
24 0nep0 4089 . . . . . . . 8 ∅ ≠ {∅}
25 0ex 4055 . . . . . . . . . . 11 ∅ ∈ V
2625, 2prss 3676 . . . . . . . . . 10 ((∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o) ↔ {∅, {∅}} ⊆ 𝒫 1o)
2717, 26mpbir 145 . . . . . . . . 9 (∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o)
28 en2eqpr 6801 . . . . . . . . . 10 ((𝒫 1o ≈ 2o ∧ ∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o) → (∅ ≠ {∅} → 𝒫 1o = {∅, {∅}}))
29283expb 1182 . . . . . . . . 9 ((𝒫 1o ≈ 2o ∧ (∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o)) → (∅ ≠ {∅} → 𝒫 1o = {∅, {∅}}))
3027, 29mpan2 421 . . . . . . . 8 (𝒫 1o ≈ 2o → (∅ ≠ {∅} → 𝒫 1o = {∅, {∅}}))
3124, 30mpi 15 . . . . . . 7 (𝒫 1o ≈ 2o → 𝒫 1o = {∅, {∅}})
3231eleq2d 2209 . . . . . 6 (𝒫 1o ≈ 2o → (𝑥 ∈ 𝒫 1o𝑥 ∈ {∅, {∅}}))
3332, 11, 133bitr3g 221 . . . . 5 (𝒫 1o ≈ 2o → (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
3433biimpd 143 . . . 4 (𝒫 1o ≈ 2o → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
3534alrimiv 1846 . . 3 (𝒫 1o ≈ 2o → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
3635, 5sylibr 133 . 2 (𝒫 1o ≈ 2oEXMID)
3723, 36impbii 125 1 (EXMID ↔ 𝒫 1o ≈ 2o)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 697  wal 1329   = wceq 1331  wcel 1480  wne 2308  Vcvv 2686  wss 3071  c0 3363  𝒫 cpw 3510  {csn 3527  {cpr 3528   class class class wbr 3929  EXMIDwem 4118  1oc1o 6306  2oc2o 6307  cen 6632
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-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-exmid 4119  df-id 4215  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1o 6313  df-2o 6314  df-en 6635
This theorem is referenced by:  pwf1oexmid  13301
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