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Mirrors > Home > ILE Home > Th. List > 3nsssucpw1 | GIF version |
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.) |
Ref | Expression |
---|---|
3nsssucpw1 | ⊢ (¬ EXMID → ¬ 3o ⊆ suc 𝒫 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3o 6416 | . . . . . 6 ⊢ 3o = suc 2o | |
2 | 1 | sseq1i 3181 | . . . . 5 ⊢ (3o ⊆ suc 𝒫 1o ↔ suc 2o ⊆ suc 𝒫 1o) |
3 | 1lt2o 6440 | . . . . . . . . 9 ⊢ 1o ∈ 2o | |
4 | ssnel 4567 | . . . . . . . . 9 ⊢ (2o ⊆ 1o → ¬ 1o ∈ 2o) | |
5 | 3, 4 | mt2 640 | . . . . . . . 8 ⊢ ¬ 2o ⊆ 1o |
6 | 2onn 6519 | . . . . . . . . . 10 ⊢ 2o ∈ ω | |
7 | 6 | elexi 2749 | . . . . . . . . 9 ⊢ 2o ∈ V |
8 | 7 | elpw 3581 | . . . . . . . 8 ⊢ (2o ∈ 𝒫 1o ↔ 2o ⊆ 1o) |
9 | 5, 8 | mtbir 671 | . . . . . . 7 ⊢ ¬ 2o ∈ 𝒫 1o |
10 | 9 | a1i 9 | . . . . . 6 ⊢ (suc 2o ⊆ suc 𝒫 1o → ¬ 2o ∈ 𝒫 1o) |
11 | sucssel 4423 | . . . . . . . . 9 ⊢ (2o ∈ ω → (suc 2o ⊆ suc 𝒫 1o → 2o ∈ suc 𝒫 1o)) | |
12 | 6, 11 | ax-mp 5 | . . . . . . . 8 ⊢ (suc 2o ⊆ suc 𝒫 1o → 2o ∈ suc 𝒫 1o) |
13 | elsuci 4402 | . . . . . . . 8 ⊢ (2o ∈ suc 𝒫 1o → (2o ∈ 𝒫 1o ∨ 2o = 𝒫 1o)) | |
14 | 12, 13 | syl 14 | . . . . . . 7 ⊢ (suc 2o ⊆ suc 𝒫 1o → (2o ∈ 𝒫 1o ∨ 2o = 𝒫 1o)) |
15 | 14 | orcomd 729 | . . . . . 6 ⊢ (suc 2o ⊆ suc 𝒫 1o → (2o = 𝒫 1o ∨ 2o ∈ 𝒫 1o)) |
16 | 10, 15 | ecased 1349 | . . . . 5 ⊢ (suc 2o ⊆ suc 𝒫 1o → 2o = 𝒫 1o) |
17 | 2, 16 | sylbi 121 | . . . 4 ⊢ (3o ⊆ suc 𝒫 1o → 2o = 𝒫 1o) |
18 | 17 | eqcomd 2183 | . . 3 ⊢ (3o ⊆ suc 𝒫 1o → 𝒫 1o = 2o) |
19 | exmidpweq 6906 | . . 3 ⊢ (EXMID ↔ 𝒫 1o = 2o) | |
20 | 18, 19 | sylibr 134 | . 2 ⊢ (3o ⊆ suc 𝒫 1o → EXMID) |
21 | 20 | con3i 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 3129 𝒫 cpw 3575 EXMIDwem 4193 suc csuc 4364 ωcom 4588 1oc1o 6407 2oc2o 6408 3oc3o 6409 |
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 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 |
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-int 3845 df-tr 4101 df-exmid 4194 df-iord 4365 df-on 4367 df-suc 4370 df-iom 4589 df-1o 6414 df-2o 6415 df-3o 6416 |
This theorem is referenced by: onntri45 7237 |
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