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Mirrors > Home > MPE Home > Th. List > Mathboxes > sdomne0 | Structured version Visualization version GIF version |
Description: A class that strictly dominates any set is not empty. (Suggested by SN, 14-Jan-2025.) (Contributed by RP, 14-Jan-2025.) |
Ref | Expression |
---|---|
sdomne0 | ⊢ (𝐵 ≺ 𝐴 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 8897 | . . . . 5 ⊢ Rel ≺ | |
2 | 1 | brrelex1i 5693 | . . . 4 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ∈ V) |
3 | breq1 5113 | . . . . . . 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 9055 | . . . . . 6 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ 𝐵 ≠ ∅)) | |
7 | sdomtr 9066 | . . . . . . 7 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≺ 𝐴) → ∅ ≺ 𝐴) | |
8 | 7 | ex 414 | . . . . . 6 ⊢ (∅ ≺ 𝐵 → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴)) |
9 | 6, 8 | syl6bir 254 | . . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ≠ ∅ → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴))) |
10 | 5, 9 | pm2.61dne 3032 | . . . 4 ⊢ (𝐵 ∈ V → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴)) |
11 | 2, 10 | syl 17 | . . 3 ⊢ (𝐵 ≺ 𝐴 → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴)) |
12 | 1 | brrelex2i 5694 | . . . . 5 ⊢ (∅ ≺ 𝐴 → 𝐴 ∈ V) |
13 | 0sdomg 9055 | . . . . 5 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (∅ ≺ 𝐴 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
15 | 14 | ibi 267 | . . 3 ⊢ (∅ ≺ 𝐴 → 𝐴 ≠ ∅) |
16 | 11, 15 | syl6 35 | . 2 ⊢ (𝐵 ≺ 𝐴 → (𝐵 ≺ 𝐴 → 𝐴 ≠ ∅)) |
17 | 16 | pm2.43i 52 | 1 ⊢ (𝐵 ≺ 𝐴 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 Vcvv 3448 ∅c0 4287 class class class wbr 5110 ≺ csdm 8889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 |
This theorem is referenced by: (None) |
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