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Mirrors > Home > MPE Home > Th. List > 0sdom1dom | Structured version Visualization version GIF version |
Description: Strict dominance over 0 is the same as dominance over 1. For a shorter proof requiring ax-un 7677, see 0sdom1domALT . (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7677. (Revised by BTernaryTau, 7-Dec-2024.) |
Ref | Expression |
---|---|
0sdom1dom | ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 8897 | . . 3 ⊢ Rel ≺ | |
2 | 1 | brrelex2i 5694 | . 2 ⊢ (∅ ≺ 𝐴 → 𝐴 ∈ V) |
3 | reldom 8896 | . . 3 ⊢ Rel ≼ | |
4 | 3 | brrelex2i 5694 | . 2 ⊢ (1o ≼ 𝐴 → 𝐴 ∈ V) |
5 | 0sdomg 9055 | . . 3 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
6 | n0 4311 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
7 | snssi 4773 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
8 | df1o2 8424 | . . . . . . . . . . 11 ⊢ 1o = {∅} | |
9 | 0ex 5269 | . . . . . . . . . . . 12 ⊢ ∅ ∈ V | |
10 | vex 3452 | . . . . . . . . . . . 12 ⊢ 𝑥 ∈ V | |
11 | en2sn 8992 | . . . . . . . . . . . 12 ⊢ ((∅ ∈ V ∧ 𝑥 ∈ V) → {∅} ≈ {𝑥}) | |
12 | 9, 10, 11 | mp2an 691 | . . . . . . . . . . 11 ⊢ {∅} ≈ {𝑥} |
13 | 8, 12 | eqbrtri 5131 | . . . . . . . . . 10 ⊢ 1o ≈ {𝑥} |
14 | endom 8926 | . . . . . . . . . 10 ⊢ (1o ≈ {𝑥} → 1o ≼ {𝑥}) | |
15 | 13, 14 | ax-mp 5 | . . . . . . . . 9 ⊢ 1o ≼ {𝑥} |
16 | domssr 8946 | . . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ {𝑥} ⊆ 𝐴 ∧ 1o ≼ {𝑥}) → 1o ≼ 𝐴) | |
17 | 15, 16 | mp3an3 1451 | . . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ {𝑥} ⊆ 𝐴) → 1o ≼ 𝐴) |
18 | 17 | ex 414 | . . . . . . 7 ⊢ (𝐴 ∈ V → ({𝑥} ⊆ 𝐴 → 1o ≼ 𝐴)) |
19 | 7, 18 | syl5 34 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 → 1o ≼ 𝐴)) |
20 | 19 | exlimdv 1937 | . . . . 5 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ 𝐴 → 1o ≼ 𝐴)) |
21 | 6, 20 | biimtrid 241 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ≠ ∅ → 1o ≼ 𝐴)) |
22 | 1n0 8439 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
23 | dom0 9053 | . . . . . . 7 ⊢ (1o ≼ ∅ ↔ 1o = ∅) | |
24 | 22, 23 | nemtbir 3041 | . . . . . 6 ⊢ ¬ 1o ≼ ∅ |
25 | breq2 5114 | . . . . . 6 ⊢ (𝐴 = ∅ → (1o ≼ 𝐴 ↔ 1o ≼ ∅)) | |
26 | 24, 25 | mtbiri 327 | . . . . 5 ⊢ (𝐴 = ∅ → ¬ 1o ≼ 𝐴) |
27 | 26 | necon2ai 2974 | . . . 4 ⊢ (1o ≼ 𝐴 → 𝐴 ≠ ∅) |
28 | 21, 27 | impbid1 224 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ 1o ≼ 𝐴)) |
29 | 5, 28 | bitrd 279 | . 2 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)) |
30 | 2, 4, 29 | pm5.21nii 380 | 1 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2944 Vcvv 3448 ⊆ wss 3915 ∅c0 4287 {csn 4591 class class class wbr 5110 1oc1o 8410 ≈ 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-pr 5389 |
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-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-sn 4592 df-pr 4594 df-op 4598 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-suc 6328 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-1o 8417 df-en 8891 df-dom 8892 df-sdom 8893 |
This theorem is referenced by: 1sdom2 9191 sdom1OLD 9194 1sdom2dom 9198 djulepw 10135 fin45 10335 gchxpidm 10612 rankcf 10720 snct 31672 |
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