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Theorem pw1nel3 7229
Description: Negated excluded middle implies that the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
Assertion
Ref Expression
pw1nel3 EXMID → ¬ 𝒫 1o ∈ 3o)

Proof of Theorem pw1nel3
StepHypRef Expression
1 pw1ne0 7226 . . . . 5 𝒫 1o ≠ ∅
2 pw1ne1 7227 . . . . 5 𝒫 1o ≠ 1o
31, 2nelpri 3616 . . . 4 ¬ 𝒫 1o ∈ {∅, 1o}
43a1i 9 . . 3 EXMID → ¬ 𝒫 1o ∈ {∅, 1o})
5 df2o3 6430 . . . 4 2o = {∅, 1o}
65eleq2i 2244 . . 3 (𝒫 1o ∈ 2o ↔ 𝒫 1o ∈ {∅, 1o})
74, 6sylnibr 677 . 2 EXMID → ¬ 𝒫 1o ∈ 2o)
8 exmidpweq 6908 . . . 4 (EXMID ↔ 𝒫 1o = 2o)
98notbii 668 . . 3 EXMID ↔ ¬ 𝒫 1o = 2o)
10 1oex 6424 . . . . . 6 1o ∈ V
1110pwex 4183 . . . . 5 𝒫 1o ∈ V
1211elsn 3608 . . . 4 (𝒫 1o ∈ {2o} ↔ 𝒫 1o = 2o)
1312notbii 668 . . 3 (¬ 𝒫 1o ∈ {2o} ↔ ¬ 𝒫 1o = 2o)
149, 13sylbb2 138 . 2 EXMID → ¬ 𝒫 1o ∈ {2o})
15 df-3o 6418 . . . . . . 7 3o = suc 2o
16 df-suc 4371 . . . . . . 7 suc 2o = (2o ∪ {2o})
1715, 16eqtri 2198 . . . . . 6 3o = (2o ∪ {2o})
1817eleq2i 2244 . . . . 5 (𝒫 1o ∈ 3o ↔ 𝒫 1o ∈ (2o ∪ {2o}))
19 elun 3276 . . . . 5 (𝒫 1o ∈ (2o ∪ {2o}) ↔ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}))
2018, 19bitri 184 . . . 4 (𝒫 1o ∈ 3o ↔ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}))
2120notbii 668 . . 3 (¬ 𝒫 1o ∈ 3o ↔ ¬ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}))
22 ioran 752 . . 3 (¬ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}) ↔ (¬ 𝒫 1o ∈ 2o ∧ ¬ 𝒫 1o ∈ {2o}))
2321, 22bitri 184 . 2 (¬ 𝒫 1o ∈ 3o ↔ (¬ 𝒫 1o ∈ 2o ∧ ¬ 𝒫 1o ∈ {2o}))
247, 14, 23sylanbrc 417 1 EXMID → ¬ 𝒫 1o ∈ 3o)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708   = wceq 1353  wcel 2148  cun 3127  c0 3422  𝒫 cpw 3575  {csn 3592  {cpr 3593  EXMIDwem 4194  suc csuc 4365  1oc1o 6409  2oc2o 6410  3oc3o 6411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4102  df-exmid 4195  df-iord 4366  df-on 4368  df-suc 4371  df-1o 6416  df-2o 6417  df-3o 6418
This theorem is referenced by:  sucpw1ne3  7230  sucpw1nss3  7233
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