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Mirrors > Home > MPE Home > Th. List > Mathboxes > sucidALT | Structured version Visualization version GIF version |
Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 43616 using translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sucidALT.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
sucidALT | ⊢ 𝐴 ∈ suc 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucidALT.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | snid 4663 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
3 | elun1 4175 | . . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 𝐴 ∈ ({𝐴} ∪ 𝐴) |
5 | df-suc 6367 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
6 | 5 | equncomi 4154 | . 2 ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴) |
7 | 4, 6 | eleqtrri 2832 | 1 ⊢ 𝐴 ∈ suc 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3474 ∪ cun 3945 {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-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-un 3952 df-in 3954 df-ss 3964 df-sn 4628 df-suc 6367 |
This theorem is referenced by: (None) |
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