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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 4716 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | 1 | biimpi 215 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
3 | 2 | uneq2d 4158 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅)) |
4 | df-suc 6363 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | un0 4385 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
6 | 5 | eqcomi 2735 | . 2 ⊢ 𝐴 = (𝐴 ∪ ∅) |
7 | 3, 4, 6 | 3eqtr4g 2791 | 1 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∪ cun 3941 ∅c0 4317 {csn 4623 suc csuc 6359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-dif 3946 df-un 3948 df-nul 4318 df-sn 4624 df-suc 6363 |
This theorem is referenced by: nsuceq0 6440 sucon 7787 ordsuc 7797 ordsucOLD 7798 sucprcreg 9595 suc11reg 9613 |
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