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Theorem 3nsssucpw1 7238
Description: Negated excluded middle implies that 3o is not a subset of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
Assertion
Ref Expression
3nsssucpw1 EXMID → ¬ 3o ⊆ suc 𝒫 1o)

Proof of Theorem 3nsssucpw1
StepHypRef Expression
1 df-3o 6422 . . . . . 6 3o = suc 2o
21sseq1i 3183 . . . . 5 (3o ⊆ suc 𝒫 1o ↔ suc 2o ⊆ suc 𝒫 1o)
3 1lt2o 6446 . . . . . . . . 9 1o ∈ 2o
4 ssnel 4570 . . . . . . . . 9 (2o ⊆ 1o → ¬ 1o ∈ 2o)
53, 4mt2 640 . . . . . . . 8 ¬ 2o ⊆ 1o
6 2onn 6525 . . . . . . . . . 10 2o ∈ ω
76elexi 2751 . . . . . . . . 9 2o ∈ V
87elpw 3583 . . . . . . . 8 (2o ∈ 𝒫 1o ↔ 2o ⊆ 1o)
95, 8mtbir 671 . . . . . . 7 ¬ 2o ∈ 𝒫 1o
109a1i 9 . . . . . 6 (suc 2o ⊆ suc 𝒫 1o → ¬ 2o ∈ 𝒫 1o)
11 sucssel 4426 . . . . . . . . 9 (2o ∈ ω → (suc 2o ⊆ suc 𝒫 1o → 2o ∈ suc 𝒫 1o))
126, 11ax-mp 5 . . . . . . . 8 (suc 2o ⊆ suc 𝒫 1o → 2o ∈ suc 𝒫 1o)
13 elsuci 4405 . . . . . . . 8 (2o ∈ suc 𝒫 1o → (2o ∈ 𝒫 1o ∨ 2o = 𝒫 1o))
1412, 13syl 14 . . . . . . 7 (suc 2o ⊆ suc 𝒫 1o → (2o ∈ 𝒫 1o ∨ 2o = 𝒫 1o))
1514orcomd 729 . . . . . 6 (suc 2o ⊆ suc 𝒫 1o → (2o = 𝒫 1o ∨ 2o ∈ 𝒫 1o))
1610, 15ecased 1349 . . . . 5 (suc 2o ⊆ suc 𝒫 1o → 2o = 𝒫 1o)
172, 16sylbi 121 . . . 4 (3o ⊆ suc 𝒫 1o → 2o = 𝒫 1o)
1817eqcomd 2183 . . 3 (3o ⊆ suc 𝒫 1o → 𝒫 1o = 2o)
19 exmidpweq 6912 . . 3 (EXMID ↔ 𝒫 1o = 2o)
2018, 19sylibr 134 . 2 (3o ⊆ suc 𝒫 1oEXMID)
2120con3i 632 1 EXMID → ¬ 3o ⊆ suc 𝒫 1o)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 708   = wceq 1353  wcel 2148  wss 3131  𝒫 cpw 3577  EXMIDwem 4196  suc csuc 4367  ωcom 4591  1oc1o 6413  2oc2o 6414  3oc3o 6415
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 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538
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 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-tr 4104  df-exmid 4197  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-1o 6420  df-2o 6421  df-3o 6422
This theorem is referenced by:  onntri45  7243
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