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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 6101 | . 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) | |
2 | elirr 8854 | . . . 4 ⊢ ¬ 𝐴 ∈ 𝐴 | |
3 | df-suc 6032 | . . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
4 | 3 | eqeq1i 2776 | . . . . . 6 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴) |
5 | ssequn2 4041 | . . . . . 6 ⊢ ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴) | |
6 | 4, 5 | sylbb2 230 | . . . . 5 ⊢ (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴) |
7 | snidg 4467 | . . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
8 | ssel2 3846 | . . . . 5 ⊢ (({𝐴} ⊆ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐴) | |
9 | 6, 7, 8 | syl2an 587 | . . . 4 ⊢ ((suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝐴) |
10 | 2, 9 | mto 189 | . . 3 ⊢ ¬ (suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V) |
11 | 10 | imnani 392 | . 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V) |
12 | 1, 11 | impbii 201 | 1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 Vcvv 3408 ∪ cun 3820 ⊆ wss 3822 {csn 4435 suc csuc 6028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 ax-reg 8849 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-sn 4436 df-pr 4438 df-suc 6032 |
This theorem is referenced by: (None) |
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