Theorem uni0 2525

Description: The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 2710 by Eric Schmidt, 4-Apr-2007.)

Assertion

Ref	Expression
uni0	|- U.(/) = (/)

Proof of Theorem uni0

Step	Hyp	Ref	Expression
1		0ss 2301	. 2 |- (/) (_ {(/)}
2		uni0b 2523	. 2 |- (U.(/) = (/) <-> (/) (_ {(/)})
3	1, 2	mpbir 190	1 |- U.(/) = (/)

Syntax hints:   = wceq 956   (_ wss 2047  (/)c0 2280  {csn 2409  U.cuni 2503

This theorem is referenced by:  unisn2 2875  unizlim 3113  unixp0 3518  fvprc 3721  funfv 3770  fvopabn 3786  1stval 4081  2ndval 4082  1st0 4083  2nd0 4084  1st2val 4095  2nd2val 4096  unifiOLD 4557  infeq5 4621  rankuni 4698  rankxplim3 4714  dffsum 6998  isumnul 7203  0opnt 7601  sn0top 7647  indistop 7648

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-uni 2504

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