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Mirrors > Home > MPE Home > Th. List > dom0 | Structured version Visualization version GIF version |
Description: A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.) |
Ref | Expression |
---|---|
dom0 | ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 8370 | . . . . 5 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 5501 | . . . 4 ⊢ (𝐴 ≼ ∅ → 𝐴 ∈ V) |
3 | 0domg 8498 | . . . 4 ⊢ (𝐴 ∈ V → ∅ ≼ 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ≼ ∅ → ∅ ≼ 𝐴) |
5 | 4 | pm4.71i 560 | . 2 ⊢ (𝐴 ≼ ∅ ↔ (𝐴 ≼ ∅ ∧ ∅ ≼ 𝐴)) |
6 | sbthb 8492 | . 2 ⊢ ((𝐴 ≼ ∅ ∧ ∅ ≼ 𝐴) ↔ 𝐴 ≈ ∅) | |
7 | en0 8427 | . 2 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
8 | 5, 6, 7 | 3bitri 298 | 1 ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 Vcvv 3440 ∅c0 4217 class class class wbr 4968 ≈ cen 8361 ≼ cdom 8362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-er 8146 df-en 8365 df-dom 8366 |
This theorem is referenced by: fin1a2lem11 9685 cfpwsdom 9859 |
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