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Mirrors > Home > ILE Home > Th. List > sucpw1ne3 | GIF version |
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.) |
Ref | Expression |
---|---|
sucpw1ne3 | ⊢ (¬ EXMID → suc 𝒫 1o ≠ 3o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw1nel3 7243 | . 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 3o) | |
2 | 1oex 6438 | . . . . . 6 ⊢ 1o ∈ V | |
3 | 2 | pwex 4195 | . . . . 5 ⊢ 𝒫 1o ∈ V |
4 | 3 | sucid 4429 | . . . 4 ⊢ 𝒫 1o ∈ suc 𝒫 1o |
5 | eleq2 2251 | . . . 4 ⊢ (suc 𝒫 1o = 3o → (𝒫 1o ∈ suc 𝒫 1o ↔ 𝒫 1o ∈ 3o)) | |
6 | 4, 5 | mpbii 148 | . . 3 ⊢ (suc 𝒫 1o = 3o → 𝒫 1o ∈ 3o) |
7 | 6 | necon3bi 2407 | . 2 ⊢ (¬ 𝒫 1o ∈ 3o → suc 𝒫 1o ≠ 3o) |
8 | 1, 7 | syl 14 | 1 ⊢ (¬ EXMID → suc 𝒫 1o ≠ 3o) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1363 ∈ wcel 2158 ≠ wne 2357 𝒫 cpw 3587 EXMIDwem 4206 suc csuc 4377 1oc1o 6423 3oc3o 6425 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-uni 3822 df-tr 4114 df-exmid 4207 df-iord 4378 df-on 4380 df-suc 4383 df-1o 6430 df-2o 6431 df-3o 6432 |
This theorem is referenced by: (None) |
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