Theorem eqimss 3051

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 3014	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
2	1	simplbi 268	1 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1284   ⊆ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  eqimss2  3052  uneqin  3215  sssnr  3545  sssnm  3546  ssprr  3548  sstpr  3549  snsspw  3556  elpwuni  3762  disjeq2  3770  disjeq1  3773  pwne  3934  pwssunim  4039  poeq2  4055  seeq1  4094  seeq2  4095  trsucss  4178  onsucelsucr  4252  xp11m  4779  funeq  4941  fnresdm  5028  fssxp  5078  ffdm  5081  fcoi1  5090  fof  5126  dff1o2  5151  fvmptss2  5268  fvmptssdm  5276  fprg  5367  dff1o6  5436  tposeq  5885  nntri1  6097  nntri2or2  6099  nnsseleq  6102  frec2uzf1od  9408