Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > eqimss | GIF version |
Description: Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Ref | Expression |
---|---|
eqimss | ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqss 2960 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
2 | 1 | simplbi 259 | 1 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ⊆ wss 2917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 |
This theorem is referenced by: eqimss2 2998 sspssr 3043 sspsstrir 3046 uneqin 3188 sssnr 3524 sssnm 3525 ssprr 3527 sstpr 3528 snsspw 3535 elpwuni 3741 disjeq2 3749 disjeq1 3752 pwne 3913 pwssunim 4021 poeq2 4037 seeq1 4076 seeq2 4077 trsucss 4160 onsucelsucr 4234 xp11m 4759 funeq 4921 fnresdm 5008 fssxp 5058 ffdm 5061 fcoi1 5070 fof 5106 dff1o2 5131 fvmptss2 5247 fvmptssdm 5255 fprg 5346 dff1o6 5416 tposeq 5862 nntri1 6074 nntri2or2 6076 nnsseleq 6079 frec2uzf1od 9192 |
Copyright terms: Public domain | W3C validator |