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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 6453 | . 2 ⊢ 𝐴 ∈ suc 𝐴 |
3 | suceq 6437 | . 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵) | |
4 | 2, 3 | eleqtrid 2831 | 1 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 suc csuc 6373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-un 3949 df-sn 4631 df-suc 6377 |
This theorem is referenced by: pssnn 9193 pssnnOLD 9290 |
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