Theorem fundm2domnop 11279

Description: A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 12-Oct-2020.) (Proof shortened by AV, 9-Jun-2021.)

Assertion

Ref	Expression
fundm2domnop	|- ( ( Fun G /\ 2o ~<_ dom G ) -> -. G e. ( _V X. _V ) )

Proof of Theorem fundm2domnop

Step	Hyp	Ref	Expression
1		fundif 5420	. 2 |- ( Fun G -> Fun ( G \ { (/) } ) )
2		fundm2domnop0 11278	. 2 |- ( ( Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> -. G e. ( _V X. _V ) )
3	1, 2	sylan 283	1 |- ( ( Fun G /\ 2o ~<_ dom G ) -> -. G e. ( _V X. _V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2209   _Vcvv 2821   \ cdif 3217   (/)c0 3520   {csn 3705   class class class wbr 4125   X. cxp 4767   dom cdm 4769   Fun wfun 5366   2oc2o 6671   ~<_ cdom 7011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fv 5380  df-1o 6677  df-2o 6678  df-dom 7014

This theorem is referenced by: (None)