![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sseq1i | GIF version |
Description: An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
sseq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
sseq1i | ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | sseq1 3178 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 ⊆ wss 3129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-in 3135 df-ss 3142 |
This theorem is referenced by: eqsstri 3187 eqsstrid 3201 ssab 3225 rabss 3232 uniiunlem 3244 prss 3747 prssg 3748 tpss 3756 iunss 3925 pwtr 4216 ordsucss 4500 elomssom 4601 cores2 5137 dffun2 5222 funimaexglem 5295 idref 5752 ordgt0ge1 6430 3nsssucpw1 7229 prarloclemn 7489 bdeqsuc 14289 bj-omssind 14343 |
Copyright terms: Public domain | W3C validator |