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Theorem exmidpw 6551
Description: Excluded middle is equivalent to the power set of 1𝑜 having two elements. Remark of [PradicBrown2021], 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 6126 . . . . 5 1𝑜 = {∅}
2 p0ex 3987 . . . . 5 {∅} ∈ V
31, 2eqeltri 2155 . . . 4 1𝑜 ∈ V
43pwex 3982 . . 3 𝒫 1𝑜 ∈ V
5 exmid01 3996 . . . . . . . . 9 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
65biimpi 118 . . . . . . . 8 (EXMID → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
7619.21bi 1491 . . . . . . 7 (EXMID → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
81pweqi 3410 . . . . . . . . 9 𝒫 1𝑜 = 𝒫 {∅}
98eleq2i 2149 . . . . . . . 8 (𝑥 ∈ 𝒫 1𝑜𝑥 ∈ 𝒫 {∅})
10 selpw 3413 . . . . . . . 8 (𝑥 ∈ 𝒫 {∅} ↔ 𝑥 ⊆ {∅})
119, 10bitri 182 . . . . . . 7 (𝑥 ∈ 𝒫 1𝑜𝑥 ⊆ {∅})
12 vex 2615 . . . . . . . 8 𝑥 ∈ V
1312elpr 3443 . . . . . . 7 (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
147, 11, 133imtr4g 203 . . . . . 6 (EXMID → (𝑥 ∈ 𝒫 1𝑜𝑥 ∈ {∅, {∅}}))
1514ssrdv 3016 . . . . 5 (EXMID → 𝒫 1𝑜 ⊆ {∅, {∅}})
16 pwpw0ss 3622 . . . . . . 7 {∅, {∅}} ⊆ 𝒫 {∅}
1716, 8sseqtr4i 3043 . . . . . 6 {∅, {∅}} ⊆ 𝒫 1𝑜
1817a1i 9 . . . . 5 (EXMID → {∅, {∅}} ⊆ 𝒫 1𝑜)
1915, 18eqssd 3027 . . . 4 (EXMID → 𝒫 1𝑜 = {∅, {∅}})
20 df2o2 6128 . . . 4 2𝑜 = {∅, {∅}}
2119, 20syl6eqr 2133 . . 3 (EXMID → 𝒫 1𝑜 = 2𝑜)
22 eqeng 6413 . . 3 (𝒫 1𝑜 ∈ V → (𝒫 1𝑜 = 2𝑜 → 𝒫 1𝑜 ≈ 2𝑜))
234, 21, 22mpsyl 64 . 2 (EXMID → 𝒫 1𝑜 ≈ 2𝑜)
24 0nep0 3965 . . . . . . . 8 ∅ ≠ {∅}
25 0ex 3931 . . . . . . . . . . 11 ∅ ∈ V
2625, 2prss 3567 . . . . . . . . . 10 ((∅ ∈ 𝒫 1𝑜 ∧ {∅} ∈ 𝒫 1𝑜) ↔ {∅, {∅}} ⊆ 𝒫 1𝑜)
2717, 26mpbir 144 . . . . . . . . 9 (∅ ∈ 𝒫 1𝑜 ∧ {∅} ∈ 𝒫 1𝑜)
28 en2eqpr 6550 . . . . . . . . . 10 ((𝒫 1𝑜 ≈ 2𝑜 ∧ ∅ ∈ 𝒫 1𝑜 ∧ {∅} ∈ 𝒫 1𝑜) → (∅ ≠ {∅} → 𝒫 1𝑜 = {∅, {∅}}))
29283expb 1140 . . . . . . . . 9 ((𝒫 1𝑜 ≈ 2𝑜 ∧ (∅ ∈ 𝒫 1𝑜 ∧ {∅} ∈ 𝒫 1𝑜)) → (∅ ≠ {∅} → 𝒫 1𝑜 = {∅, {∅}}))
3027, 29mpan2 416 . . . . . . . 8 (𝒫 1𝑜 ≈ 2𝑜 → (∅ ≠ {∅} → 𝒫 1𝑜 = {∅, {∅}}))
3124, 30mpi 15 . . . . . . 7 (𝒫 1𝑜 ≈ 2𝑜 → 𝒫 1𝑜 = {∅, {∅}})
3231eleq2d 2152 . . . . . 6 (𝒫 1𝑜 ≈ 2𝑜 → (𝑥 ∈ 𝒫 1𝑜𝑥 ∈ {∅, {∅}}))
3332, 11, 133bitr3g 220 . . . . 5 (𝒫 1𝑜 ≈ 2𝑜 → (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
3433biimpd 142 . . . 4 (𝒫 1𝑜 ≈ 2𝑜 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
3534alrimiv 1797 . . 3 (𝒫 1𝑜 ≈ 2𝑜 → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
3635, 5sylibr 132 . 2 (𝒫 1𝑜 ≈ 2𝑜EXMID)
3723, 36impbii 124 1 (EXMID ↔ 𝒫 1𝑜 ≈ 2𝑜)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  wal 1283   = wceq 1285  wcel 1434  wne 2249  Vcvv 2612  wss 2984  c0 3269  𝒫 cpw 3406  {csn 3422  {cpr 3423   class class class wbr 3811  EXMIDwem 3993  1𝑜c1o 6106  2𝑜c2o 6107  cen 6385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-exmid 3994  df-id 4084  df-suc 4162  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-1o 6113  df-2o 6114  df-en 6388
This theorem is referenced by: (None)
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