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Mirrors > Home > ILE Home > Th. List > pw1ne3 | GIF version |
Description: The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
Ref | Expression |
---|---|
pw1ne3 | ⊢ 𝒫 1o ≠ 3o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2o 6438 | . . . . 5 ⊢ 1o ∈ 2o | |
2 | ssnel 4566 | . . . . 5 ⊢ (2o ⊆ 1o → ¬ 1o ∈ 2o) | |
3 | 1, 2 | mt2 640 | . . . 4 ⊢ ¬ 2o ⊆ 1o |
4 | 2onn 6517 | . . . . . 6 ⊢ 2o ∈ ω | |
5 | 4 | elexi 2749 | . . . . 5 ⊢ 2o ∈ V |
6 | 5 | elpw 3581 | . . . 4 ⊢ (2o ∈ 𝒫 1o ↔ 2o ⊆ 1o) |
7 | 3, 6 | mtbir 671 | . . 3 ⊢ ¬ 2o ∈ 𝒫 1o |
8 | 5 | sucid 4415 | . . . . 5 ⊢ 2o ∈ suc 2o |
9 | df-3o 6414 | . . . . 5 ⊢ 3o = suc 2o | |
10 | 8, 9 | eleqtrri 2253 | . . . 4 ⊢ 2o ∈ 3o |
11 | eleq2 2241 | . . . 4 ⊢ (𝒫 1o = 3o → (2o ∈ 𝒫 1o ↔ 2o ∈ 3o)) | |
12 | 10, 11 | mpbiri 168 | . . 3 ⊢ (𝒫 1o = 3o → 2o ∈ 𝒫 1o) |
13 | 7, 12 | mto 662 | . 2 ⊢ ¬ 𝒫 1o = 3o |
14 | 13 | neir 2350 | 1 ⊢ 𝒫 1o ≠ 3o |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ⊆ wss 3129 𝒫 cpw 3575 suc csuc 4363 ωcom 4587 1oc1o 6405 2oc2o 6406 3oc3o 6407 |
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 4119 ax-nul 4127 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-setind 4534 |
This theorem depends on definitions: df-bi 117 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 3809 df-int 3844 df-tr 4100 df-iord 4364 df-on 4366 df-suc 4369 df-iom 4588 df-1o 6412 df-2o 6413 df-3o 6414 |
This theorem is referenced by: 3nelsucpw1 7228 |
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