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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 7249 | . 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 3o) | |
2 | 1oex 6443 | . . . . . 6 ⊢ 1o ∈ V | |
3 | 2 | pwex 4198 | . . . . 5 ⊢ 𝒫 1o ∈ V |
4 | 3 | sucid 4432 | . . . 4 ⊢ 𝒫 1o ∈ suc 𝒫 1o |
5 | eleq2 2253 | . . . 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 2410 | . 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 1364 ∈ wcel 2160 ≠ wne 2360 𝒫 cpw 3590 EXMIDwem 4209 suc csuc 4380 1oc1o 6428 3oc3o 6430 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-tr 4117 df-exmid 4210 df-iord 4381 df-on 4383 df-suc 4386 df-1o 6435 df-2o 6436 df-3o 6437 |
This theorem is referenced by: (None) |
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