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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 5721 | . 2 ⊢ 𝐴 ∈ suc 𝐴 |
3 | suceq 5707 | . 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵) | |
4 | 2, 3 | syl5eleq 2694 | 1 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-sn 4126 df-suc 5646 |
This theorem is referenced by: pssnn 8063 |
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