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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 44890 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 4662 | . . 3 ⊢ 𝐴 ∈ {𝐴} | 
| 3 | elun1 4182 | . . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴)) | |
| 4 | 2, 3 | ax-mp 5 | . 2 ⊢ 𝐴 ∈ ({𝐴} ∪ 𝐴) | 
| 5 | df-suc 6390 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
| 6 | 5 | equncomi 4160 | . 2 ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴) | 
| 7 | 4, 6 | eleqtrri 2840 | 1 ⊢ 𝐴 ∈ suc 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {csn 4626 suc csuc 6386 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-suc 6390 | 
| This theorem is referenced by: (None) | 
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