Theorem wel 2201

Description: Extend wff definition to include atomic formulas with the membership predicate. This is read either " x is an element of y", or " x is a member of y", or " x belongs to y", or " y contains x". Note: The phrase " y includes x " means " x is a subset of y"; to use it also for x e. y, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT).

This syntactical construction introduces a binary non-logical predicate symbol e. into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for e. apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments.

Instead of introducing wel 2201 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 2200. This lets us avoid overloading the e. connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 2201 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 2200. Note: To see the proof steps of this syntax proof, type "MM> SHOW PROOF wel / ALL" in the Metamath program. (Contributed by NM, 24-Jan-2006.)

Assertion

Ref	Expression
wel	wff x e. y

Proof of Theorem wel

Step	Hyp	Ref	Expression
1		wcel 2200	1 wff x e. y

Syntax hints:   e. wcel 2200

This theorem is referenced by:  elequ1  2204  elequ2  2205  cleljust  2206  elsb1  2207  elsb2  2208  dveel1  2209  dveel2  2210  axext3  2212  axext4  2213  bm1.1  2214  ru  3027  nfuni  3894  nfunid  3895  unieq  3897  inteq  3926  nfint  3933  uniiun  4019  intiin  4020  trint  4197  axsep2  4203  bm1.3ii  4205  zfnuleu  4208  0ex  4211  nalset  4214  vnex  4215  repizf2  4246  axpweq  4255  zfpow  4259  axpow2  4260  axpow3  4261  el  4262  vpwex  4263  dtruarb  4275  exmidn0m  4285  exmidsssn  4286  fr0  4442  wetrep  4451  zfun  4525  axun2  4526  uniuni  4542  regexmid  4627  zfregfr  4666  ordwe  4668  wessep  4670  nnregexmid  4713  rele  4852  funimaexglem  5404  acexmidlem2  6004  acexmid  6006  dfsmo2  6439  smores2  6446  tfrcllemsucaccv  6506  pw2f1odclem  7003  findcard2d  7061  exmidfodomr  7393  acfun  7400  exmidontriimlem3  7416  exmidontriimlem4  7417  exmidontriim  7418  onntri13  7434  exmidontri  7435  onntri51  7436  onntri3or  7441  exmidmotap  7458  ccfunen  7461  cc1  7462  ltsopi  7518  fnn0nninf  10672  fsum2dlemstep  11961  fprod2dlemstep  12149  exmidunben  13013  prdsex  13318  isbasis3g  14736  tgcl  14754  tgss2  14769  blbas  15123  metrest  15196  dvmptfsum  15415  uhgrfm  15889  ushgrfm  15890  uhgrss  15891  uhgreq12g  15892  uhgrfun  15893  ushgruhgr  15896  isuhgropm  15897  uhgr0e  15898  uhgr0vb  15900  uhgr0  15901  uhgrun  15902  bdcuni  16322  bdcint  16323  bdcriota  16329  bdsep1  16331  bdsep2  16332  bdsepnft  16333  bdsepnf  16334  bdsepnfALT  16335  bdzfauscl  16336  bdbm1.3ii  16337  bj-axemptylem  16338  bj-axempty  16339  bj-axempty2  16340  bj-nalset  16341  bdinex1  16345  bj-zfpair2  16356  bj-axun2  16361  bj-uniex2  16362  bj-d0clsepcl  16371  bj-nn0suc0  16396  bj-nntrans  16397  bj-omex2  16423  strcollnft  16430  sscoll2  16434  nninfsellemcl  16465  nninfsellemsuc  16466  nninfsellemqall  16469  nninfomni  16473  exmidsbthrlem  16478