0sdomg - Metamath Proof Explorer

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Theorem 0sdomg 9108

Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5334, ax-un 7740. (Revised by BTernaryTau, 29-Nov-2024.)

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

**Proof of Theorem 0sdomg**
Step	Hyp	Ref	Expression
1		0domg 9106	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)
2		brsdom 8984	. . . 4 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴))
3	2	baib 545	. . 3 ⊢ (∅ ≼ 𝐴 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴))
4	1, 3	syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴))
5		en0r 9030	. . 3 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)
6	5	necon3bbii 3004	. 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 2145 ≠ wne 2957 ∅c0 4282 class class class wbr 5107 ≈ cen 8953 ≼ cdom 8954 ≺ csdm 8955

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 2147 ax-9 2155 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402

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 2566 df-clab 2741 df-cleq 2754 df-clel 2837 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-br 5108 df-opab 5172 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-en 8957 df-dom 8958 df-sdom 8959

This theorem is used by: 0sdom 9110 fodomr 9130 pwdom 9131 0sdom1dom 9220 1sdom2dom 9228 infn0ALT 9277 fodomfir 9301 fodomfib 9302 domwdom 9550 iunfictbso 10121 djulepw 10199 fin45 10398 fodomb 10533 brdom3 10535 gchxpidm 10682 inar1 10788 csdfil 24126 ovoliunnul 25741 carsgclctunlem3 34839 domalom 38166 ovoliunnfl 38419 voliunnfl 38421 volsupnfl 38422 sdomne0 44261 sdomne0d 44262 ensucne0OLD 44378 nnfoctb 45890 caragenunicl 47360

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