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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0heALT | Structured version Visualization version GIF version |
Description: The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
0heALT | ⊢ ∅ hereditary 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xphe 43353 | . 2 ⊢ (∅ × 𝐴) hereditary 𝐴 | |
2 | 0xp 5776 | . . 3 ⊢ (∅ × 𝐴) = ∅ | |
3 | heeq1 43349 | . . 3 ⊢ ((∅ × 𝐴) = ∅ → ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴) |
5 | 1, 4 | mpbi 229 | 1 ⊢ ∅ hereditary 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∅c0 4322 × cxp 5676 hereditary whe 43344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-he 43345 |
This theorem is referenced by: (None) |
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