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Theorem sucpw1ne3 7250
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 7249 . 2 EXMID → ¬ 𝒫 1o ∈ 3o)
2 1oex 6443 . . . . . 6 1o ∈ V
32pwex 4198 . . . . 5 𝒫 1o ∈ V
43sucid 4432 . . . 4 𝒫 1o ∈ suc 𝒫 1o
5 eleq2 2253 . . . 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 2410 . 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 1364  wcel 2160  wne 2360  𝒫 cpw 3590  EXMIDwem 4209  suc csuc 4380  1oc1o 6428  3oc3o 6430
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-tr 4117  df-exmid 4210  df-iord 4381  df-on 4383  df-suc 4386  df-1o 6435  df-2o 6436  df-3o 6437
This theorem is referenced by: (None)
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