![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 6421 | . . . . 5 ⊢ 1o ∈ 2o | |
2 | ssnel 4553 | . . . . 5 ⊢ (2o ⊆ 1o → ¬ 1o ∈ 2o) | |
3 | 1, 2 | mt2 635 | . . . 4 ⊢ ¬ 2o ⊆ 1o |
4 | 2onn 6500 | . . . . . 6 ⊢ 2o ∈ ω | |
5 | 4 | elexi 2742 | . . . . 5 ⊢ 2o ∈ V |
6 | 5 | elpw 3572 | . . . 4 ⊢ (2o ∈ 𝒫 1o ↔ 2o ⊆ 1o) |
7 | 3, 6 | mtbir 666 | . . 3 ⊢ ¬ 2o ∈ 𝒫 1o |
8 | 5 | sucid 4402 | . . . . 5 ⊢ 2o ∈ suc 2o |
9 | df-3o 6397 | . . . . 5 ⊢ 3o = suc 2o | |
10 | 8, 9 | eleqtrri 2246 | . . . 4 ⊢ 2o ∈ 3o |
11 | eleq2 2234 | . . . 4 ⊢ (𝒫 1o = 3o → (2o ∈ 𝒫 1o ↔ 2o ∈ 3o)) | |
12 | 10, 11 | mpbiri 167 | . . 3 ⊢ (𝒫 1o = 3o → 2o ∈ 𝒫 1o) |
13 | 7, 12 | mto 657 | . 2 ⊢ ¬ 𝒫 1o = 3o |
14 | 13 | neir 2343 | 1 ⊢ 𝒫 1o ≠ 3o |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ⊆ wss 3121 𝒫 cpw 3566 suc csuc 4350 ωcom 4574 1oc1o 6388 2oc2o 6389 3oc3o 6390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-1o 6395 df-2o 6396 df-3o 6397 |
This theorem is referenced by: 3nelsucpw1 7211 |
Copyright terms: Public domain | W3C validator |