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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 7738, see 0sdom1domALT . (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7738. (Revised by BTernaryTau, 7-Dec-2024.) |
Ref | Expression |
---|---|
0sdom1dom | ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 8969 | . . 3 ⊢ Rel ≺ | |
2 | 1 | brrelex2i 5729 | . 2 ⊢ (∅ ≺ 𝐴 → 𝐴 ∈ V) |
3 | reldom 8968 | . . 3 ⊢ Rel ≼ | |
4 | 3 | brrelex2i 5729 | . 2 ⊢ (1o ≼ 𝐴 → 𝐴 ∈ V) |
5 | 0sdomg 9127 | . . 3 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
6 | n0 4342 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
7 | snssi 4807 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
8 | df1o2 8492 | . . . . . . . . . . 11 ⊢ 1o = {∅} | |
9 | 0ex 5302 | . . . . . . . . . . . 12 ⊢ ∅ ∈ V | |
10 | vex 3467 | . . . . . . . . . . . 12 ⊢ 𝑥 ∈ V | |
11 | en2sn 9064 | . . . . . . . . . . . 12 ⊢ ((∅ ∈ V ∧ 𝑥 ∈ V) → {∅} ≈ {𝑥}) | |
12 | 9, 10, 11 | mp2an 690 | . . . . . . . . . . 11 ⊢ {∅} ≈ {𝑥} |
13 | 8, 12 | eqbrtri 5164 | . . . . . . . . . 10 ⊢ 1o ≈ {𝑥} |
14 | endom 8998 | . . . . . . . . . 10 ⊢ (1o ≈ {𝑥} → 1o ≼ {𝑥}) | |
15 | 13, 14 | ax-mp 5 | . . . . . . . . 9 ⊢ 1o ≼ {𝑥} |
16 | domssr 9018 | . . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ {𝑥} ⊆ 𝐴 ∧ 1o ≼ {𝑥}) → 1o ≼ 𝐴) | |
17 | 15, 16 | mp3an3 1446 | . . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ {𝑥} ⊆ 𝐴) → 1o ≼ 𝐴) |
18 | 17 | ex 411 | . . . . . . 7 ⊢ (𝐴 ∈ V → ({𝑥} ⊆ 𝐴 → 1o ≼ 𝐴)) |
19 | 7, 18 | syl5 34 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 → 1o ≼ 𝐴)) |
20 | 19 | exlimdv 1928 | . . . . 5 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ 𝐴 → 1o ≼ 𝐴)) |
21 | 6, 20 | biimtrid 241 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ≠ ∅ → 1o ≼ 𝐴)) |
22 | 1n0 8507 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
23 | dom0 9125 | . . . . . . 7 ⊢ (1o ≼ ∅ ↔ 1o = ∅) | |
24 | 22, 23 | nemtbir 3028 | . . . . . 6 ⊢ ¬ 1o ≼ ∅ |
25 | breq2 5147 | . . . . . 6 ⊢ (𝐴 = ∅ → (1o ≼ 𝐴 ↔ 1o ≼ ∅)) | |
26 | 24, 25 | mtbiri 326 | . . . . 5 ⊢ (𝐴 = ∅ → ¬ 1o ≼ 𝐴) |
27 | 26 | necon2ai 2960 | . . . 4 ⊢ (1o ≼ 𝐴 → 𝐴 ≠ ∅) |
28 | 21, 27 | impbid1 224 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ 1o ≼ 𝐴)) |
29 | 5, 28 | bitrd 278 | . 2 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)) |
30 | 2, 4, 29 | pm5.21nii 377 | 1 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2930 Vcvv 3463 ⊆ wss 3939 ∅c0 4318 {csn 4624 class class class wbr 5143 1oc1o 8478 ≈ cen 8959 ≼ cdom 8960 ≺ csdm 8961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-suc 6370 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-1o 8485 df-en 8963 df-dom 8964 df-sdom 8965 |
This theorem is referenced by: 1sdom2 9263 sdom1OLD 9266 1sdom2dom 9270 djulepw 10215 fin45 10415 gchxpidm 10692 rankcf 10800 snct 32540 |
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