Theorem eqimss 2997

Description: Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Assertion

Ref	Expression
eqimss	⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)

Proof of Theorem eqimss

Step	Hyp	Ref	Expression
1		eqss 2960	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
2	1	simplbi 259	1 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)

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