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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 6462 | . 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) | |
2 | elirr 9635 | . . . 4 ⊢ ¬ 𝐴 ∈ 𝐴 | |
3 | df-suc 6392 | . . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
4 | 3 | eqeq1i 2740 | . . . . . 6 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴) |
5 | ssequn2 4199 | . . . . . 6 ⊢ ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴) | |
6 | 4, 5 | sylbb2 238 | . . . . 5 ⊢ (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴) |
7 | snidg 4665 | . . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
8 | ssel2 3990 | . . . . 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 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ⊆ wss 3963 {csn 4631 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-pr 5438 ax-reg 9630 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-sn 4632 df-pr 4634 df-suc 6392 |
This theorem is referenced by: (None) |
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