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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 6269 is sucidVD 41204 without virtual deductions and was automatically
derived from sucidVD 41204.
|
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 4600 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
3 | elun2 4152 | . . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ (𝐴 ∪ {𝐴})) | |
4 | 2, 3 | e0a 41104 | . 2 ⊢ 𝐴 ∈ (𝐴 ∪ {𝐴}) |
5 | df-suc 6196 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
6 | 4, 5 | eleqtrri 2912 | 1 ⊢ 𝐴 ∈ suc 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Vcvv 3494 ∪ cun 3933 {csn 4566 suc csuc 6192 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3940 df-in 3942 df-ss 3951 df-sn 4567 df-suc 6196 |
This theorem is referenced by: (None) |
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