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Mirrors > Home > MPE Home > Th. List > 0sdomg | Structured version Visualization version GIF version |
Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) |
Ref | Expression |
---|---|
0sdomg | ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0domg 8638 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) | |
2 | brsdom 8526 | . . . 4 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴)) | |
3 | 2 | baib 538 | . . 3 ⊢ (∅ ≼ 𝐴 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
5 | ensymb 8551 | . . . 4 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 ≈ ∅) | |
6 | en0 8566 | . . . 4 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
7 | 5, 6 | bitri 277 | . . 3 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅) |
8 | 7 | necon3bbii 3063 | . 2 ⊢ (¬ ∅ ≈ 𝐴 ↔ 𝐴 ≠ ∅) |
9 | 4, 8 | syl6bb 289 | 1 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∅c0 4290 class class class wbr 5058 ≈ cen 8500 ≼ cdom 8501 ≺ csdm 8502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 |
This theorem is referenced by: 0sdom 8642 fodomr 8662 pwdom 8663 sdom1 8712 infn0 8774 fodomfib 8792 domwdom 9032 iunfictbso 9534 djulepw 9612 fin45 9808 fodomb 9942 brdom3 9944 gchxpidm 10085 inar1 10191 csdfil 22496 ovoliunnul 24102 carsgclctunlem3 31573 domalom 34679 ovoliunnfl 34928 voliunnfl 34930 volsupnfl 34931 ensucne0OLD 39889 nnfoctb 41302 caragenunicl 42800 |
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