0sdomg - Metamath Proof Explorer

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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 5330, ax-un 7736. (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 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 2145 ≠ wne 2955 ∅c0 4279 class class class wbr 5103 ≈ 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 2147 ax-9 2155 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pr 5398

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 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-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5168 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 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 10529 brdom3 10531 gchxpidm 10678 inar1 10784 csdfil 24120 ovoliunnul 25735 carsgclctunlem3 34831 domalom 38158 ovoliunnfl 38411 voliunnfl 38413 volsupnfl 38414 sdomne0 44253 sdomne0d 44254 ensucne0OLD 44370 nnfoctb 45882 caragenunicl 47352

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