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Mirrors > Home > MPE Home > Th. List > Mathboxes > idhe | Structured version Visualization version GIF version |
Description: The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.) |
Ref | Expression |
---|---|
idhe | ⊢ I hereditary 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idssxp 5915 | . 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) | |
2 | dfhe2 40118 | . 2 ⊢ ( I hereditary 𝐴 ↔ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)) | |
3 | 1, 2 | mpbir 233 | 1 ⊢ I hereditary 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3935 I cid 5458 × cxp 5552 ↾ cres 5556 hereditary whe 40116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-he 40117 |
This theorem is referenced by: sshepw 40133 |
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