Theorem imadisjlnd 38459

Description: Natural deduction form of one negated side of imadisj 5484. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypothesis

Ref	Expression
imadisjlnd.1	|- ( ph -> ( dom A i^i B ) =/= (/) )

Assertion

Ref	Expression
imadisjlnd	|- ( ph -> ( A " B ) =/= (/) )

Proof of Theorem imadisjlnd

Step	Hyp	Ref	Expression
1		imadisjlnd.1	. 2 |- ( ph -> ( dom A i^i B ) =/= (/) )
2		imadisj 5484	. . . 4 |- ( ( A " B ) = (/) <-> ( dom A i^i B ) = (/) )
3	2	biimpi 206	. . 3 |- ( ( A " B ) = (/) -> ( dom A i^i B ) = (/) )
4	3	necon3i 2826	. 2 |- ( ( dom A i^i B ) =/= (/) -> ( A " B ) =/= (/) )
5	1, 4	syl 17	1 |- ( ph -> ( A " B ) =/= (/) )

Syntax hints:   -> wi 4   = wceq 1483   =/= wne 2794   i^i cin 3573   (/)c0 3915   dom cdm 5114   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by: (None)