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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 8991 | . . . . 5 ⊢ Rel ≺ | |
2 | 1 | brrelex1i 5745 | . . . 4 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ∈ V) |
3 | breq1 5151 | . . . . . . 7 ⊢ (𝐵 = ∅ → (𝐵 ≺ 𝐴 ↔ ∅ ≺ 𝐴)) | |
4 | 3 | biimpd 229 | . . . . . 6 ⊢ (𝐵 = ∅ → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴)) |
5 | 4 | a1i 11 | . . . . 5 ⊢ (𝐵 ∈ V → (𝐵 = ∅ → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴))) |
6 | 0sdomg 9143 | . . . . . 6 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ 𝐵 ≠ ∅)) | |
7 | sdomtr 9154 | . . . . . . 7 ⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≺ 𝐴) → ∅ ≺ 𝐴) | |
8 | 7 | ex 412 | . . . . . 6 ⊢ (∅ ≺ 𝐵 → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴)) |
9 | 6, 8 | biimtrrdi 254 | . . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ≠ ∅ → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴))) |
10 | 5, 9 | pm2.61dne 3026 | . . . 4 ⊢ (𝐵 ∈ V → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴)) |
11 | 2, 10 | syl 17 | . . 3 ⊢ (𝐵 ≺ 𝐴 → (𝐵 ≺ 𝐴 → ∅ ≺ 𝐴)) |
12 | 1 | brrelex2i 5746 | . . . . 5 ⊢ (∅ ≺ 𝐴 → 𝐴 ∈ V) |
13 | 0sdomg 9143 | . . . . 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 206 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 class class class wbr 5148 ≺ csdm 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 |
This theorem is referenced by: (None) |
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