0sdomg - Metamath Proof Explorer

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Theorem 0sdomg 9128

Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5326, ax-un 7751. (Revised by BTernaryTau, 29-Nov-2024.)

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

**Proof of Theorem 0sdomg**
Step	Hyp	Ref	Expression
1		0domg 9126	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)
2		brsdom 9004	. . . 4 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴))
3	2	baib 545	. . 3 ⊢ (∅ ≼ 𝐴 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴))
4	1, 3	syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴))
5		en0r 9050	. . 3 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)
6	5	necon3bbii 3002	. 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 2144 ≠ wne 2955 ∅c0 4278 class class class wbr 5102 ≈ cen 8973 ≼ cdom 8974 ≺ csdm 8975

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-sb 2099 df-mo 2564 df-clab 2739 df-cleq 2752 df-clel 2835 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-br 5103 df-opab 5167 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-en 8977 df-dom 8978 df-sdom 8979

This theorem is used by: 0sdom 9130 fodomr 9150 pwdom 9151 0sdom1dom 9240 1sdom2dom 9248 infn0ALT 9298 fodomfir 9322 fodomfib 9323 domwdom 9571 iunfictbso 10196 djulepw 10274 fin45 10473 fodomb 10608 brdom3 10610 gchxpidm 10757 inar1 10863 csdfil 24218 ovoliunnul 25833 carsgclctunlem3 34998 domalom 38420 ovoliunnfl 38673 voliunnfl 38675 volsupnfl 38676 sdomne0 44507 sdomne0d 44508 ensucne0OLD 44624 nnfoctb 46158 caragenunicl 47627

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