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Mirrors > Home > MPE Home > Th. List > sdom1OLD | Structured version Visualization version GIF version |
Description: Obsolete version of sdom1 9276 as of 12-Dec-2024. (Contributed by Stefan O'Rear, 28-Oct-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sdom1OLD | ⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domnsym 9138 | . . . . 5 ⊢ (1o ≼ 𝐴 → ¬ 𝐴 ≺ 1o) | |
2 | 1 | con2i 139 | . . . 4 ⊢ (𝐴 ≺ 1o → ¬ 1o ≼ 𝐴) |
3 | 0sdom1dom 9272 | . . . 4 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) | |
4 | 2, 3 | sylnibr 329 | . . 3 ⊢ (𝐴 ≺ 1o → ¬ ∅ ≺ 𝐴) |
5 | relsdom 8991 | . . . . 5 ⊢ Rel ≺ | |
6 | 5 | brrelex1i 5745 | . . . 4 ⊢ (𝐴 ≺ 1o → 𝐴 ∈ V) |
7 | 0sdomg 9143 | . . . . 5 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
8 | 7 | necon2bbid 2982 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 = ∅ ↔ ¬ ∅ ≺ 𝐴)) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (𝐴 ≺ 1o → (𝐴 = ∅ ↔ ¬ ∅ ≺ 𝐴)) |
10 | 4, 9 | mpbird 257 | . 2 ⊢ (𝐴 ≺ 1o → 𝐴 = ∅) |
11 | 1n0 8525 | . . . 4 ⊢ 1o ≠ ∅ | |
12 | 1oex 8515 | . . . . 5 ⊢ 1o ∈ V | |
13 | 12 | 0sdom 9146 | . . . 4 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅) |
14 | 11, 13 | mpbir 231 | . . 3 ⊢ ∅ ≺ 1o |
15 | breq1 5151 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 1o ↔ ∅ ≺ 1o)) | |
16 | 14, 15 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≺ 1o) |
17 | 10, 16 | impbii 209 | 1 ⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 class class class wbr 5148 1oc1o 8498 ≼ cdom 8982 ≺ 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-suc 6392 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 |
This theorem is referenced by: (None) |
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