Theorem fundm2domnop 11109

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 5374	. 2 |- ( Fun G -> Fun ( G \ { (/) } ) )
2		fundm2domnop0 11108	. 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 2202   _Vcvv 2802   \ cdif 3197   (/)c0 3494   {csn 3669   class class class wbr 4088   X. cxp 4723   dom cdm 4725   Fun wfun 5320   2oc2o 6575   ~<_ cdom 6907

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fv 5334  df-1o 6581  df-2o 6582  df-dom 6910

This theorem is referenced by: (None)