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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 43617 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 4664 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
3 | elun1 4176 | . . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 𝐴 ∈ ({𝐴} ∪ 𝐴) |
5 | df-suc 6368 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
6 | 5 | equncomi 4155 | . 2 ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴) |
7 | 4, 6 | eleqtrri 2833 | 1 ⊢ 𝐴 ∈ suc 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3475 ∪ cun 3946 {csn 4628 suc csuc 6364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-un 3953 df-in 3955 df-ss 3965 df-sn 4629 df-suc 6368 |
This theorem is referenced by: (None) |
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