Theorem nn0eln0 4657

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 4656	. 2 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )
2		noel 3455	. . . . 5 |- -. (/) e. (/)
3		eleq2 2260	. . . . 5 |- ( A = (/) -> ( (/) e. A <-> (/) e. (/) ) )
4	2, 3	mtbiri 676	. . . 4 |- ( A = (/) -> -. (/) e. A )
5		nner 2371	. . . 4 |- ( A = (/) -> -. A =/= (/) )
6	4, 5	2falsed 703	. . 3 |- ( A = (/) -> ( (/) e. A <-> A =/= (/) ) )
7		id 19	. . . 4 |- ( (/) e. A -> (/) e. A )
8		ne0i 3458	. . . 4 |- ( (/) e. A -> A =/= (/) )
9	7, 8	2thd 175	. . 3 |- ( (/) e. A -> ( (/) e. A <-> A =/= (/) ) )
10	6, 9	jaoi 717	. 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 709   = wceq 1364   e. wcel 2167   =/= wne 2367   (/)c0 3451   omcom 4627

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-uni 3841  df-int 3876  df-suc 4407  df-iom 4628

This theorem is referenced by:  nnmord  6584  bj-charfunr  15540