Theorem infcl 8394

Description: An infimum belongs to its base class (closure law). See also inflb 8395 and infglb 8396. (Contributed by AV, 3-Sep-2020.)

Hypotheses

Ref	Expression
infcl.1	|- ( ph -> R Or A )
infcl.2	|- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )

Assertion

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

Distinct variable groups:   x, A, y, z   x, B, y, z   x, R, y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem infcl

Step	Hyp	Ref	Expression
1		df-inf 8349	. 2 |- inf ( B , A , R ) = sup ( B , A , `' R )
2		infcl.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		infcl.2	. . . 4 |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )
6	2, 5	infcllem 8393	. . 3 |- ( ph -> E. x e. A ( A. y e. B -. x `' R y /\ A. y e. A ( y `' R x -> E. z e. B y `' R z ) ) )
7	4, 6	supcl 8364	. 2 |- ( ph -> sup ( B , A , `' R ) e. A )
8	1, 7	syl5eqel 2705	1 |- ( ph -> inf ( B , A , R ) e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   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-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-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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  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-iota 5851  df-riota 6611  df-sup 8348  df-inf 8349

This theorem is referenced by:  infrecl  11005  infxrcl  12163  infssd  29488  xrge0infssd  29526  infxrge0lb  29529  infxrge0gelb  29531  omsf  30358  wzel  31771  wsuccl  31776