Theorem snhesn 38080

Description: Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)

Assertion

Ref	Expression
snhesn	|- { <. A , A >. } hereditary { B }

Proof of Theorem snhesn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . . 7 |- x e. _V
2	1	elima3 5473	. . . . . 6 |- ( x e. ( { <. A , A >. } " { B } ) <-> E. y ( y e. { B } /\ <. y , x >. e. { <. A , A >. } ) )
3		velsn 4193	. . . . . 6 |- ( x e. { B } <-> x = B )
4	2, 3	imbi12i 340	. . . . 5 |- ( ( x e. ( { <. A , A >. } " { B } ) -> x e. { B } ) <-> ( E. y ( y e. { B } /\ <. y , x >. e. { <. A , A >. } ) -> x = B ) )
5	4	albii 1747	. . . 4 |- ( A. x ( x e. ( { <. A , A >. } " { B } ) -> x e. { B } ) <-> A. x ( E. y ( y e. { B } /\ <. y , x >. e. { <. A , A >. } ) -> x = B ) )
6		velsn 4193	. . . . . . . 8 |- ( y e. { B } <-> y = B )
7		opex 4932	. . . . . . . . . 10 |- <. y , x >. e. _V
8	7	elsn 4192	. . . . . . . . 9 |- ( <. y , x >. e. { <. A , A >. } <-> <. y , x >. = <. A , A >. )
9		vex 3203	. . . . . . . . . 10 |- y e. _V
10	9, 1	opth 4945	. . . . . . . . 9 |- ( <. y , x >. = <. A , A >. <-> ( y = A /\ x = A ) )
11	8, 10	bitri 264	. . . . . . . 8 |- ( <. y , x >. e. { <. A , A >. } <-> ( y = A /\ x = A ) )
12	6, 11	anbi12i 733	. . . . . . 7 |- ( ( y e. { B } /\ <. y , x >. e. { <. A , A >. } ) <-> ( y = B /\ ( y = A /\ x = A ) ) )
13		3anass 1042	. . . . . . 7 |- ( ( y = B /\ y = A /\ x = A ) <-> ( y = B /\ ( y = A /\ x = A ) ) )
14	12, 13	bitr4i 267	. . . . . 6 |- ( ( y e. { B } /\ <. y , x >. e. { <. A , A >. } ) <-> ( y = B /\ y = A /\ x = A ) )
15		simp3 1063	. . . . . . 7 |- ( ( y = B /\ y = A /\ x = A ) -> x = A )
16		simp2 1062	. . . . . . 7 |- ( ( y = B /\ y = A /\ x = A ) -> y = A )
17		simp1 1061	. . . . . . 7 |- ( ( y = B /\ y = A /\ x = A ) -> y = B )
18	15, 16, 17	3eqtr2d 2662	. . . . . 6 |- ( ( y = B /\ y = A /\ x = A ) -> x = B )
19	14, 18	sylbi 207	. . . . 5 |- ( ( y e. { B } /\ <. y , x >. e. { <. A , A >. } ) -> x = B )
20	19	exlimiv 1858	. . . 4 |- ( E. y ( y e. { B } /\ <. y , x >. e. { <. A , A >. } ) -> x = B )
21	5, 20	mpgbir 1726	. . 3 |- A. x ( x e. ( { <. A , A >. } " { B } ) -> x e. { B } )
22		dfss2 3591	. . 3 |- ( ( { <. A , A >. } " { B } ) C_ { B } <-> A. x ( x e. ( { <. A , A >. } " { B } ) -> x e. { B } ) )
23	21, 22	mpbir 221	. 2 |- ( { <. A , A >. } " { B } ) C_ { B }
24		df-he 38067	. 2 |- ( { <. A , A >. } hereditary { B } <-> ( { <. A , A >. } " { B } ) C_ { B } )
25	23, 24	mpbir 221	1 |- { <. A , A >. } hereditary { B }

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   C_ wss 3574   {csn 4177   <.cop 4183   "cima 5117   hereditary whe 38066

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-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  df-he 38067

This theorem is referenced by: (None)