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Mirrors > Home > ILE Home > Th. List > 0domg | GIF version |
Description: Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
0domg | ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3325 | . 2 ⊢ ∅ ⊆ 𝐴 | |
2 | ssdomg 6549 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ⊆ 𝐴 → ∅ ≼ 𝐴)) | |
3 | 1, 2 | mpi 15 | 1 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1439 ⊆ wss 3000 ∅c0 3287 class class class wbr 3851 ≼ cdom 6510 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-dom 6513 |
This theorem is referenced by: 0dom 6609 |
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