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Mirrors > Home > MPE Home > Th. List > Mathboxes > sucidVD | Structured version Visualization version GIF version |
Description: A set belongs to its successor. The following User's Proof is a
Virtual Deduction proof completed automatically by the tools
program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2
and Norm Megill's Metamath Proof Assistant.
sucid 5721 is sucidVD 38130 without virtual deductions and was automatically
derived from sucidVD 38130.
|
Ref | Expression |
---|---|
sucidVD.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
sucidVD | ⊢ 𝐴 ∈ suc 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucidVD.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | snid 4155 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
3 | elun2 3743 | . . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ (𝐴 ∪ {𝐴})) | |
4 | 2, 3 | e0a 38020 | . 2 ⊢ 𝐴 ∈ (𝐴 ∪ {𝐴}) |
5 | df-suc 5646 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
6 | 4, 5 | eleqtrri 2687 | 1 ⊢ 𝐴 ∈ suc 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 {csn 4125 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-sn 4126 df-suc 5646 |
This theorem is referenced by: (None) |
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