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  5998  acexmid  6000  dfsmo2  6433  smores2  6440  tfrcllemsucaccv  6500  pw2f1odclem  6995  findcard2d  7053  exmidfodomr  7382  acfun  7389  exmidontriimlem3  7405  exmidontriimlem4  7406  exmidontriim  7407  onntri13  7423  exmidontri  7424  onntri51  7425  onntri3or  7430  exmidmotap  7447  ccfunen  7450  cc1  7451  ltsopi  7507  fnn0nninf  10660  fsum2dlemstep  11945  fprod2dlemstep  12133  exmidunben  12997  prdsex  13302  isbasis3g  14720  tgcl  14738  tgss2  14753  blbas  15107  metrest  15180  dvmptfsum  15399  uhgrfm  15873  ushgrfm  15874  uhgrss  15875  uhgreq12g  15876  uhgrfun  15877  ushgruhgr  15880  isuhgropm  15881  uhgr0e  15882  uhgr0vb  15884  uhgr0  15885  uhgrun  15886  bdcuni  16239  bdcint  16240  bdcriota  16246  bdsep1  16248  bdsep2  16249  bdsepnft  16250  bdsepnf  16251  bdsepnfALT  16252  bdzfauscl  16253  bdbm1.3ii  16254  bj-axemptylem  16255  bj-axempty  16256  bj-axempty2  16257  bj-nalset  16258  bdinex1  16262  bj-zfpair2  16273  bj-axun2  16278  bj-uniex2  16279  bj-d0clsepcl  16288  bj-nn0suc0  16313  bj-nntrans  16314  bj-omex2  16340  strcollnft  16347  sscoll2  16351  nninfsellemcl  16377  nninfsellemsuc  16378  nninfsellemqall  16381  nninfomni  16385  exmidsbthrlem  16390