Theorem nn0eln0 4744

Description: A natural number is nonempty iff it contains the empty set. Although in constructive mathematics it is generally more natural to work with inhabited sets and ignore the whole concept of nonempty sets, in the specific case of natural numbers this theorem may be helpful in converting proofs which were written assuming excluded middle. (Contributed by Jim Kingdon, 28-Aug-2019.)

Assertion

Ref	Expression
nn0eln0	|- ( A e. om -> ( (/) e. A <-> A =/= (/) ) )

Proof of Theorem nn0eln0

Step	Hyp	Ref	Expression
1		0elnn 4743	. 2 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )
2		noel 3514	. . . . 5 |- -. (/) e. (/)
3		eleq2 2298	. . . . 5 |- ( A = (/) -> ( (/) e. A <-> (/) e. (/) ) )
4	2, 3	mtbiri 682	. . . 4 |- ( A = (/) -> -. (/) e. A )
5		nner 2418	. . . 4 |- ( A = (/) -> -. A =/= (/) )
6	4, 5	2falsed 710	. . 3 |- ( A = (/) -> ( (/) e. A <-> A =/= (/) ) )
7		id 19	. . . 4 |- ( (/) e. A -> (/) e. A )
8		ne0i 3517	. . . 4 |- ( (/) e. A -> A =/= (/) )
9	7, 8	2thd 175	. . 3 |- ( (/) e. A -> ( (/) e. A <-> A =/= (/) ) )
10	6, 9	jaoi 724	. 2 |- ( ( A = (/) \/ (/) e. A ) -> ( (/) e. A <-> A =/= (/) ) )
11	1, 10	syl 14	1 |- ( A e. om -> ( (/) e. A <-> A =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   =/= wne 2414   (/)c0 3510   omcom 4714

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-uni 3917  df-int 3952  df-suc 4494  df-iom 4715

This theorem is referenced by:  nnmord  6752  bj-charfunr  16597