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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 8250 | . . . . . 6 ⊢ Rel ≺ | |
2 | 1 | brrelex1i 5408 | . . . . 5 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V) |
3 | ssdif0 4172 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∖ 𝐴) = ∅) | |
4 | ssdomg 8289 | . . . . . . 7 ⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) | |
5 | domnsym 8376 | . . . . . . 7 ⊢ (𝐵 ≼ 𝐴 → ¬ 𝐴 ≺ 𝐵) | |
6 | 4, 5 | syl6 35 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → ¬ 𝐴 ≺ 𝐵)) |
7 | 3, 6 | syl5bir 235 | . . . . 5 ⊢ (𝐴 ∈ V → ((𝐵 ∖ 𝐴) = ∅ → ¬ 𝐴 ≺ 𝐵)) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝐴 ≺ 𝐵 → ((𝐵 ∖ 𝐴) = ∅ → ¬ 𝐴 ≺ 𝐵)) |
9 | 8 | con2d 132 | . . 3 ⊢ (𝐴 ≺ 𝐵 → (𝐴 ≺ 𝐵 → ¬ (𝐵 ∖ 𝐴) = ∅)) |
10 | 9 | pm2.43i 52 | . 2 ⊢ (𝐴 ≺ 𝐵 → ¬ (𝐵 ∖ 𝐴) = ∅) |
11 | 10 | neqned 2976 | 1 ⊢ (𝐴 ≺ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 Vcvv 3398 ∖ cdif 3789 ⊆ wss 3792 ∅c0 4141 class class class wbr 4888 ≼ cdom 8241 ≺ csdm 8242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 |
This theorem is referenced by: domtriomlem 9601 konigthlem 9727 odcau 18407 |
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