![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0domgOLD | Structured version Visualization version GIF version |
Description: Obsolete version of 0domg 9131 as of 29-Nov-2024. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0domgOLD | ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4400 | . 2 ⊢ ∅ ⊆ 𝐴 | |
2 | ssdomg 9027 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ⊆ 𝐴 → ∅ ≼ 𝐴)) | |
3 | 1, 2 | mpi 20 | 1 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ⊆ wss 3949 ∅c0 4326 class class class wbr 5152 ≼ cdom 8968 |
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-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-dom 8972 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |