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Mathbox for Richard Penner |
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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 5132 |
. . . 4
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2 | relssdmrn 5356 |
. . . 4
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3 | 1, 2 | ax-mp 5 |
. . 3
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4 | dmresi 5160 |
. . . . 5
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5 | 4 | eqimssi 3486 |
. . . 4
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6 | rnresi 5181 |
. . . . 5
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7 | 6 | eqimssi 3486 |
. . . 4
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8 | xpss12 4940 |
. . . 4
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9 | 5, 7, 8 | mp2an 678 |
. . 3
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10 | 3, 9 | sstri 3441 |
. 2
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11 | dfhe2 36369 |
. 2
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12 | 10, 11 | mpbir 213 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pr 4639 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-sn 3969 df-pr 3971 df-op 3975 df-br 4403 df-opab 4462 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-he 36368 |
This theorem is referenced by: sshepw 36385 |
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