0sdomg - Metamath Proof Explorer

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Theorem 0sdomg 9019

Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5303, ax-un 7668. (Revised by BTernaryTau, 29-Nov-2024.)

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

**Proof of Theorem 0sdomg**
Step	Hyp	Ref	Expression
1		0domg 9017	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)
2		brsdom 8897	. . . 4 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴))
3	2	baib 535	. . 3 ⊢ (∅ ≼ 𝐴 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴))
4	1, 3	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴))
5		en0r 8942	. . 3 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)
6	5	necon3bbii 2975	. 2 ⊢ (¬ ∅ ≈ 𝐴 ↔ 𝐴 ≠ ∅)
7	4, 6	bitrdi 287	1 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∈ wcel 2111 ≠ wne 2928 ∅c0 4283 class class class wbr 5091 ≈ cen 8866 ≼ cdom 8867 ≺ csdm 8868

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pr 5370

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-mo 2535 df-clab 2710 df-cleq 2723 df-clel 2806 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3905 df-un 3907 df-ss 3919 df-nul 4284 df-if 4476 df-sn 4577 df-pr 4579 df-op 4583 df-br 5092 df-opab 5154 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-en 8870 df-dom 8871 df-sdom 8872

This theorem is referenced by: 0sdom 9021 fodomr 9041 pwdom 9042 0sdom1dom 9130 1sdom2dom 9138 infn0ALT 9187 fodomfir 9212 fodomfib 9213 fodomfibOLD 9215 domwdom 9460 iunfictbso 10002 djulepw 10081 fin45 10280 fodomb 10414 brdom3 10416 gchxpidm 10557 inar1 10663 csdfil 23807 ovoliunnul 25433 carsgclctunlem3 34328 domalom 37437 ovoliunnfl 37701 voliunnfl 37703 volsupnfl 37704 sdomne0 43445 sdomne0d 43446 ensucne0OLD 43562 nnfoctb 45084 caragenunicl 46561

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