Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > eqelsuc | 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 4339 | . 2 ⊢ 𝐴 ∈ suc 𝐴 |
3 | suceq 4324 | . 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵) | |
4 | 2, 3 | eleqtrid 2228 | 1 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 suc csuc 4287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-suc 4293 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |