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Mirrors > Home > MPE Home > Th. List > sdom0OLD | Structured version Visualization version GIF version |
Description: Obsolete version of sdom0 9124 as of 29-Nov-2024. (Contributed by NM, 26-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sdom0OLD | ⊢ ¬ 𝐴 ≺ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 8962 | . . . 4 ⊢ Rel ≺ | |
2 | 1 | brrelex1i 5728 | . . 3 ⊢ (𝐴 ≺ ∅ → 𝐴 ∈ V) |
3 | 0domg 9116 | . . 3 ⊢ (𝐴 ∈ V → ∅ ≼ 𝐴) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ≺ ∅ → ∅ ≼ 𝐴) |
5 | domnsym 9115 | . . 3 ⊢ (∅ ≼ 𝐴 → ¬ 𝐴 ≺ ∅) | |
6 | 5 | con2i 139 | . 2 ⊢ (𝐴 ≺ ∅ → ¬ ∅ ≼ 𝐴) |
7 | 4, 6 | pm2.65i 193 | 1 ⊢ ¬ 𝐴 ≺ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2099 Vcvv 3469 ∅c0 4318 class class class wbr 5142 ≼ cdom 8953 ≺ csdm 8954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 |
This theorem is referenced by: (None) |
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