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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 6468 | . 2 ⊢ 𝐴 ∈ suc 𝐴 |
3 | suceq 6452 | . 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵) | |
4 | 2, 3 | eleqtrid 2845 | 1 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-sn 4632 df-suc 6392 |
This theorem is referenced by: pssnn 9207 |
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