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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 6437 | . 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) | |
2 | elirr 9588 | . . . 4 ⊢ ¬ 𝐴 ∈ 𝐴 | |
3 | df-suc 6367 | . . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
4 | 3 | eqeq1i 2737 | . . . . . 6 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴) |
5 | ssequn2 4182 | . . . . . 6 ⊢ ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴) | |
6 | 4, 5 | sylbb2 237 | . . . . 5 ⊢ (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴) |
7 | snidg 4661 | . . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
8 | ssel2 3976 | . . . . 5 ⊢ (({𝐴} ⊆ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐴) | |
9 | 6, 7, 8 | syl2an 596 | . . . 4 ⊢ ((suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝐴) |
10 | 2, 9 | mto 196 | . . 3 ⊢ ¬ (suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V) |
11 | 10 | imnani 401 | . 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 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∪ cun 3945 ⊆ wss 3947 {csn 4627 suc csuc 6363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-pr 5426 ax-reg 9583 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-sn 4628 df-pr 4630 df-suc 6367 |
This theorem is referenced by: (None) |
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