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Mirrors > Home > MPE Home > Th. List > Mathboxes > idhe | Structured version Visualization version Unicode version |
Description: The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.) |
Ref | Expression |
---|---|
idhe | hereditary |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 5426 | . . . 4 | |
2 | relssdmrn 5656 | . . . 4 | |
3 | 1, 2 | ax-mp 5 | . . 3 |
4 | dmresi 5457 | . . . . 5 | |
5 | 4 | eqimssi 3659 | . . . 4 |
6 | rnresi 5479 | . . . . 5 | |
7 | 6 | eqimssi 3659 | . . . 4 |
8 | xpss12 5225 | . . . 4 | |
9 | 5, 7, 8 | mp2an 708 | . . 3 |
10 | 3, 9 | sstri 3612 | . 2 |
11 | dfhe2 38068 | . 2 hereditary | |
12 | 10, 11 | mpbir 221 | 1 hereditary |
Colors of variables: wff setvar class |
Syntax hints: wss 3574 cid 5023 cxp 5112 cdm 5114 crn 5115 cres 5116 wrel 5119 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-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-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-he 38067 |
This theorem is referenced by: sshepw 38083 |
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