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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 215 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
3 | 2 | uneq2d 4159 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅)) |
4 | df-suc 6369 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | un0 4386 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
6 | 5 | eqcomi 2736 | . 2 ⊢ 𝐴 = (𝐴 ∪ ∅) |
7 | 3, 4, 6 | 3eqtr4g 2792 | 1 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∪ cun 3942 ∅c0 4318 {csn 4624 suc csuc 6365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-v 3471 df-dif 3947 df-un 3949 df-nul 4319 df-sn 4625 df-suc 6369 |
This theorem is referenced by: nsuceq0 6446 sucon 7800 ordsuc 7810 ordsucOLD 7811 sucprcreg 9616 suc11reg 9634 |
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