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Mirrors > Home > MPE Home > Th. List > 0sdomgOLD | Structured version Visualization version GIF version |
Description: Obsolete version of 0sdomg 9055 as of 29-Nov-2024. (Contributed by NM, 23-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0sdomgOLD | ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0domg 9051 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) | |
2 | brsdom 8922 | . . . 4 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴)) | |
3 | 2 | baib 537 | . . 3 ⊢ (∅ ≼ 𝐴 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
5 | ensymb 8949 | . . . 4 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 ≈ ∅) | |
6 | en0 8964 | . . . 4 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
7 | 5, 6 | bitri 275 | . . 3 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅) |
8 | 7 | necon3bbii 2992 | . 2 ⊢ (¬ ∅ ≈ 𝐴 ↔ 𝐴 ≠ ∅) |
9 | 4, 8 | bitrdi 287 | 1 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∅c0 4287 class class class wbr 5110 ≈ cen 8887 ≼ cdom 8888 ≺ 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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