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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 8130 | . . . . . 6 ⊢ Rel ≺ | |
2 | 1 | brrelexi 5315 | . . . . 5 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V) |
3 | ssdif0 4085 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∖ 𝐴) = ∅) | |
4 | ssdomg 8169 | . . . . . . 7 ⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) | |
5 | domnsym 8253 | . . . . . . 7 ⊢ (𝐵 ≼ 𝐴 → ¬ 𝐴 ≺ 𝐵) | |
6 | 4, 5 | syl6 35 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → ¬ 𝐴 ≺ 𝐵)) |
7 | 3, 6 | syl5bir 233 | . . . . 5 ⊢ (𝐴 ∈ V → ((𝐵 ∖ 𝐴) = ∅ → ¬ 𝐴 ≺ 𝐵)) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝐴 ≺ 𝐵 → ((𝐵 ∖ 𝐴) = ∅ → ¬ 𝐴 ≺ 𝐵)) |
9 | 8 | con2d 129 | . . 3 ⊢ (𝐴 ≺ 𝐵 → (𝐴 ≺ 𝐵 → ¬ (𝐵 ∖ 𝐴) = ∅)) |
10 | 9 | pm2.43i 52 | . 2 ⊢ (𝐴 ≺ 𝐵 → ¬ (𝐵 ∖ 𝐴) = ∅) |
11 | 10 | neqned 2939 | 1 ⊢ (𝐴 ≺ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 Vcvv 3340 ∖ cdif 3712 ⊆ wss 3715 ∅c0 4058 class class class wbr 4804 ≼ cdom 8121 ≺ csdm 8122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 |
This theorem is referenced by: domtriomlem 9476 konigthlem 9602 odcau 18239 |
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