Theorem 1div0 10233

Description: You can't divide by zero, because division explicitly excludes zero from the domain of the function. Thus, by the definition of function value, it evaluates to the empty set. (This theorem is for information only and normally is not referenced by other proofs. To be meaningful, it assumes that is not a complex number, which depends on the particular complex number construction that is used.) (Contributed by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)

Assertion

Ref	Expression
1div0

Proof of Theorem 1div0

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-div 10232	. . 3
2		riotaex 6261	. . 3
3	1, 2	dmmpt2 6870	. 2
4		eqid 2457	. . 3
5		eldifsni 4156	. . . . 5
6	5	adantl 466	. . . 4
7	6	necon2bi 2694	. . 3
8	4, 7	ax-mp 5	. 2
9		ndmovg 6458	. 2
10	3, 8, 9	mp2an 672	1

Syntax hints:  -.wn 3  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  \cdif 3472  c0 3784  {csn 4029  X.cxp 5002  domcdm 5004  iota_crio 6256  (class class class)co 6296  cc 9511  0cc0 9513  1c1 9514  cmul 9518  cdiv 10231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-riota 6257  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6800  df-2nd 6801  df-div 10232