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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 6460 | . 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) | |
| 2 | elirr 9637 | . . . 4 ⊢ ¬ 𝐴 ∈ 𝐴 | |
| 3 | df-suc 6390 | . . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
| 4 | 3 | eqeq1i 2742 | . . . . . 6 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴) |
| 5 | ssequn2 4189 | . . . . . 6 ⊢ ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴) | |
| 6 | 4, 5 | sylbb2 238 | . . . . 5 ⊢ (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴) |
| 7 | snidg 4660 | . . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
| 8 | ssel2 3978 | . . . . 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 197 | . . 3 ⊢ ¬ (suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V) |
| 11 | 10 | imnani 400 | . 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V) |
| 12 | 1, 11 | impbii 209 | 1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 {csn 4626 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-reg 9632 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-suc 6390 |
| This theorem is referenced by: (None) |
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