![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sdomdif | Structured version Visualization version GIF version |
Description: The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.) |
Ref | Expression |
---|---|
sdomdif | ⊢ (𝐴 ≺ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 8499 | . . . . . 6 ⊢ Rel ≺ | |
2 | 1 | brrelex1i 5572 | . . . . 5 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V) |
3 | ssdif0 4277 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∖ 𝐴) = ∅) | |
4 | ssdomg 8538 | . . . . . . 7 ⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) | |
5 | domnsym 8627 | . . . . . . 7 ⊢ (𝐵 ≼ 𝐴 → ¬ 𝐴 ≺ 𝐵) | |
6 | 4, 5 | syl6 35 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → ¬ 𝐴 ≺ 𝐵)) |
7 | 3, 6 | syl5bir 246 | . . . . 5 ⊢ (𝐴 ∈ V → ((𝐵 ∖ 𝐴) = ∅ → ¬ 𝐴 ≺ 𝐵)) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝐴 ≺ 𝐵 → ((𝐵 ∖ 𝐴) = ∅ → ¬ 𝐴 ≺ 𝐵)) |
9 | 8 | con2d 136 | . . 3 ⊢ (𝐴 ≺ 𝐵 → (𝐴 ≺ 𝐵 → ¬ (𝐵 ∖ 𝐴) = ∅)) |
10 | 9 | pm2.43i 52 | . 2 ⊢ (𝐴 ≺ 𝐵 → ¬ (𝐵 ∖ 𝐴) = ∅) |
11 | 10 | neqned 2994 | 1 ⊢ (𝐴 ≺ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 ≼ cdom 8490 ≺ csdm 8491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 |
This theorem is referenced by: domtriomlem 9853 konigthlem 9979 odcau 18721 |
Copyright terms: Public domain | W3C validator |