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Mirrors > Home > ILE Home > Th. List > sucprc | GIF version |
Description: A proper class is its own successor. (Contributed by NM, 3-Apr-1995.) |
Ref | Expression |
---|---|
sucprc | ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 4108 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | snprc 3435 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
3 | uneq2 3091 | . . . 4 ⊢ ({𝐴} = ∅ → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅)) | |
4 | 2, 3 | sylbi 114 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅)) |
5 | 1, 4 | syl5eq 2084 | . 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = (𝐴 ∪ ∅)) |
6 | un0 3251 | . 2 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
7 | 5, 6 | syl6eq 2088 | 1 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∪ cun 2915 ∅c0 3224 {csn 3375 suc csuc 4102 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-un 2922 df-nul 3225 df-sn 3381 df-suc 4108 |
This theorem is referenced by: sucprcreg 4273 sucon 4277 |
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