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| Mirrors > Home > MPE Home > Th. List > eqelsuc | Structured version Visualization version GIF version | ||
| Description: A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.) |
| Ref | Expression |
|---|---|
| eqelsuc.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| eqelsuc | ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqelsuc.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 2 | 1 | sucid 6466 | . 2 ⊢ 𝐴 ∈ suc 𝐴 |
| 3 | suceq 6450 | . 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵) | |
| 4 | 2, 3 | eleqtrid 2847 | 1 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 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-sn 4627 df-suc 6390 |
| This theorem is referenced by: pssnn 9208 |
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