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Theorem sucpw1ne3 7244
Description: Negated excluded middle implies that the successor of the power set of 1o is not three . (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
Assertion
Ref Expression
sucpw1ne3 EXMID → suc 𝒫 1o ≠ 3o)

Proof of Theorem sucpw1ne3
StepHypRef Expression
1 pw1nel3 7243 . 2 EXMID → ¬ 𝒫 1o ∈ 3o)
2 1oex 6438 . . . . . 6 1o ∈ V
32pwex 4195 . . . . 5 𝒫 1o ∈ V
43sucid 4429 . . . 4 𝒫 1o ∈ suc 𝒫 1o
5 eleq2 2251 . . . 4 (suc 𝒫 1o = 3o → (𝒫 1o ∈ suc 𝒫 1o ↔ 𝒫 1o ∈ 3o))
64, 5mpbii 148 . . 3 (suc 𝒫 1o = 3o → 𝒫 1o ∈ 3o)
76necon3bi 2407 . 2 (¬ 𝒫 1o ∈ 3o → suc 𝒫 1o ≠ 3o)
81, 7syl 14 1 EXMID → suc 𝒫 1o ≠ 3o)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1363  wcel 2158  wne 2357  𝒫 cpw 3587  EXMIDwem 4206  suc csuc 4377  1oc1o 6423  3oc3o 6425
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-uni 3822  df-tr 4114  df-exmid 4207  df-iord 4378  df-on 4380  df-suc 4383  df-1o 6430  df-2o 6431  df-3o 6432
This theorem is referenced by: (None)
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