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| Mirrors > Home > MPE Home > Th. List > sucprc | Structured version Visualization version 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 | snprc 4717 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 2 | 1 | biimpi 216 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
| 3 | 2 | uneq2d 4168 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅)) |
| 4 | df-suc 6390 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
| 5 | un0 4394 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
| 6 | 5 | eqcomi 2746 | . 2 ⊢ 𝐴 = (𝐴 ∪ ∅) |
| 7 | 3, 4, 6 | 3eqtr4g 2802 | 1 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∅c0 4333 {csn 4626 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-suc 6390 |
| This theorem is referenced by: nsuceq0 6467 sucon 7823 ordsuc 7833 ordsucOLD 7834 sucprcreg 9641 suc11reg 9659 |
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