Theorem infexd 8389

Description: An infimum is a set. (Contributed by AV, 2-Sep-2020.)

Hypothesis

Ref	Expression
infexd.1	|- ( ph -> R Or A )

Assertion

Ref	Expression
infexd	|- ( ph -> inf ( B , A , R ) e. _V )

Proof of Theorem infexd

Step	Hyp	Ref	Expression
1		df-inf 8349	. 2 |- inf ( B , A , R ) = sup ( B , A , `' R )
2		infexd.1	. . . 4 |- ( ph -> R Or A )
3		cnvso 5674	. . . 4 |- ( R Or A <-> `' R Or A )
4	2, 3	sylib 208	. . 3 |- ( ph -> `' R Or A )
5	4	supexd 8359	. 2 |- ( ph -> sup ( B , A , `' R ) e. _V )
6	1, 5	syl5eqel 2705	1 |- ( ph -> inf ( B , A , R ) e. _V )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   Or wor 5034   `'ccnv 5113   supcsup 8346  infcinf 8347

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-8 1992  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  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-rmo 2920  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-uni 4437  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-cnv 5122  df-sup 8348  df-inf 8349

This theorem is referenced by:  infex  8399  omsfval  30356  wsucex  31775  prmdvdsfmtnof1  41499