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Mirrors > Home > MPE Home > Th. List > sucprcreg | Structured version Visualization version GIF version |
Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.) (Proof shortened by BJ, 16-Apr-2019.) |
Ref | Expression |
---|---|
sucprcreg | ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucprc 6440 | . 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) | |
2 | elirr 9620 | . . . 4 ⊢ ¬ 𝐴 ∈ 𝐴 | |
3 | df-suc 6370 | . . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
4 | 3 | eqeq1i 2730 | . . . . . 6 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴) |
5 | ssequn2 4177 | . . . . . 6 ⊢ ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴) | |
6 | 4, 5 | sylbb2 237 | . . . . 5 ⊢ (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴) |
7 | snidg 4658 | . . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
8 | ssel2 3967 | . . . . 5 ⊢ (({𝐴} ⊆ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐴) | |
9 | 6, 7, 8 | syl2an 594 | . . . 4 ⊢ ((suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝐴) |
10 | 2, 9 | mto 196 | . . 3 ⊢ ¬ (suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V) |
11 | 10 | imnani 399 | . 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V) |
12 | 1, 11 | impbii 208 | 1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∪ cun 3937 ⊆ wss 3939 {csn 4624 suc csuc 6366 |
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-10 2129 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-pr 5423 ax-reg 9615 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-sn 4625 df-pr 4627 df-suc 6370 |
This theorem is referenced by: (None) |
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