Theorem eqimss2i 4042

Description: Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)

Hypothesis

Ref	Expression
eqimssi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
eqimss2i	⊢ 𝐵 ⊆ 𝐴

Proof of Theorem eqimss2i

Step	Hyp	Ref	Expression
1		ssid 4003	. 2 ⊢ 𝐵 ⊆ 𝐵
2		eqimssi.1	. 2 ⊢ 𝐴 = 𝐵
3	1, 2	sseqtrri 4018	1 ⊢ 𝐵 ⊆ 𝐴

Syntax hints:   = wceq 1539   ⊆ wss 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ss 3964

This theorem is referenced by:  cotr3  14929  supcvg  15806  prodfclim1  15843  ef0lem  16026  1strbas  17165  1strbasOLD  17166  restid  17383  cayley  19323  gsumval3  19816  gsumzaddlem  19830  kgencn3  23282  hmeores  23495  opnfbas  23566  tsmsf1o  23869  ust0  23944  icchmeo  24685  icchmeoOLD  24686  plyeq0lem  25959  ulmdvlem1  26148  basellem7  26827  basellem9  26829  dchrisumlem3  27230  structvtxvallem  28547  struct2griedg  28555  gsumhashmul  32478  cycpmfvlem  32541  cycpmfv3  32544  ivthALT  35523  aomclem4  42101  hashnzfzclim  43383  binomcxplemrat  43411  climsuselem1  44621  gsumfsupp  46858