Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
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 37095 | . 2 ⊢ (∅ × 𝐴) hereditary 𝐴 | |
2 | 0xp 5122 | . . 3 ⊢ (∅ × 𝐴) = ∅ | |
3 | heeq1 37091 | . . 3 ⊢ ((∅ × 𝐴) = ∅ → ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((∅ × 𝐴) hereditary 𝐴 ↔ ∅ hereditary 𝐴) |
5 | 1, 4 | mpbi 219 | 1 ⊢ ∅ hereditary 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∅c0 3874 × cxp 5036 hereditary whe 37086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-he 37087 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |