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Mirrors > Home > MPE Home > Th. List > Mathboxes > sdomne0d | Structured version Visualization version GIF version |
Description: A class that strictly dominates any set is not empty. (Contributed by RP, 3-Sep-2024.) |
Ref | Expression |
---|---|
sdomne0d.a | ⊢ (𝜑 → 𝐵 ≺ 𝐴) |
sdomne0d.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
Ref | Expression |
---|---|
sdomne0d | ⊢ (𝜑 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomne0d.a | . 2 ⊢ (𝜑 → 𝐵 ≺ 𝐴) | |
2 | sdomne0d.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | breq1 5150 | . . . . . . 7 ⊢ (𝐵 = ∅ → (𝐵 ≺ 𝐴 ↔ ∅ ≺ 𝐴)) | |
4 | 3 | biimpd 228 | . . . . . 6 ⊢ (𝐵 = ∅ → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴)) |
5 | 4 | a1i 11 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝐵 = ∅ → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴))) |
6 | 0sdomg 9100 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (∅ ≺ 𝐵 ↔ 𝐵 ≠ ∅)) | |
7 | sdomtr 9111 | . . . . . . 7 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≺ 𝐴) → ∅ ≺ 𝐴) | |
8 | 7 | ex 413 | . . . . . 6 ⊢ (∅ ≺ 𝐵 → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴)) |
9 | 6, 8 | syl6bir 253 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ≠ ∅ → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴))) |
10 | 5, 9 | pm2.61dne 3028 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴)) |
11 | 2, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴)) |
12 | relsdom 8942 | . . . . . 6 ⊢ Rel ≺ | |
13 | 12 | brrelex2i 5731 | . . . . 5 ⊢ (∅ ≺ 𝐴 → 𝐴 ∈ V) |
14 | 0sdomg 9100 | . . . . 5 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (∅ ≺ 𝐴 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
16 | 15 | ibi 266 | . . 3 ⊢ (∅ ≺ 𝐴 → 𝐴 ≠ ∅) |
17 | 11, 16 | syl6 35 | . 2 ⊢ (𝜑 → (𝐵 ≺ 𝐴 → 𝐴 ≠ ∅)) |
18 | 1, 17 | mpd 15 | 1 ⊢ (𝜑 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ∅c0 4321 class class class wbr 5147 ≺ csdm 8934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 |
This theorem is referenced by: safesnsupfiss 42151 safesnsupfilb 42154 |
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