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 7848 | . . . . . 6 ⊢ Rel ≺ | |
2 | 1 | brrelexi 5082 | . . . . 5 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V) |
3 | ssdif0 3896 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∖ 𝐴) = ∅) | |
4 | ssdomg 7887 | . . . . . . 7 ⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) | |
5 | domnsym 7971 | . . . . . . 7 ⊢ (𝐵 ≼ 𝐴 → ¬ 𝐴 ≺ 𝐵) | |
6 | 4, 5 | syl6 34 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → ¬ 𝐴 ≺ 𝐵)) |
7 | 3, 6 | syl5bir 232 | . . . . 5 ⊢ (𝐴 ∈ V → ((𝐵 ∖ 𝐴) = ∅ → ¬ 𝐴 ≺ 𝐵)) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝐴 ≺ 𝐵 → ((𝐵 ∖ 𝐴) = ∅ → ¬ 𝐴 ≺ 𝐵)) |
9 | 8 | con2d 128 | . . 3 ⊢ (𝐴 ≺ 𝐵 → (𝐴 ≺ 𝐵 → ¬ (𝐵 ∖ 𝐴) = ∅)) |
10 | 9 | pm2.43i 50 | . 2 ⊢ (𝐴 ≺ 𝐵 → ¬ (𝐵 ∖ 𝐴) = ∅) |
11 | 10 | neqned 2789 | 1 ⊢ (𝐴 ≺ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ≼ cdom 7839 ≺ csdm 7840 |
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-8 1979 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-pow 4769 ax-pr 4833 ax-un 6847 |
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-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: domtriomlem 9147 konigthlem 9269 odcau 17842 |
Copyright terms: Public domain | W3C validator |