0sdomg - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 0sdomg		Structured version Visualization version GIF version

Theorem 0sdomg 9104

Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5341, ax-un 7745. (Revised by BTernaryTau, 29-Nov-2024.)

**Assertion**
Ref	Expression
0sdomg	⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))

**Proof of Theorem 0sdomg**
Step	Hyp	Ref	Expression
1		0domg 9102	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)
2		brsdom 8980	. . . 4 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴))
3	2	baib 545	. . 3 ⊢ (∅ ≼ 𝐴 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴))
4	1, 3	syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴))
5		en0r 9026	. . 3 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)
6	5	necon3bbii 3008	. 2 ⊢ (¬ ∅ ≈ 𝐴 ↔ 𝐴 ≠ ∅)
7	4, 6	bitrdi 290	1 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∈ wcel 2146 ≠ wne 2961 ∅c0 4289 class class class wbr 5114 ≈ cen 8949 ≼ cdom 8950 ≺ csdm 8951

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pr 5409

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-mo 2570 df-clab 2745 df-cleq 2758 df-clel 2841 df-ne 2962 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-br 5115 df-opab 5179 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-en 8953 df-dom 8954 df-sdom 8955

This theorem is used by: 0sdom 9106 fodomr 9126 pwdom 9127 0sdom1dom 9216 1sdom2dom 9224 infn0ALT 9273 fodomfir 9297 fodomfib 9298 domwdom 9546 iunfictbso 10117 djulepw 10195 fin45 10394 fodomb 10528 brdom3 10530 gchxpidm 10672 inar1 10778 csdfil 24088 ovoliunnul 25703 carsgclctunlem3 34742 domalom 38091 ovoliunnfl 38354 voliunnfl 38356 volsupnfl 38357 sdomne0 44180 sdomne0d 44181 ensucne0OLD 44297 nnfoctb 45809 caragenunicl 47279

W3C validator