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Mirrors > Home > MPE Home > Th. List > cnv0 | Structured version Visualization version GIF version |
Description: The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 4709, ax-nul 4717, ax-pr 4833. (Revised by KP, 25-Oct-2021.) |
Ref | Expression |
---|---|
cnv0 | ⊢ ◡∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0 3888 | . 2 ⊢ (◡∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ ◡∅) | |
2 | br0 4631 | . . . . . 6 ⊢ ¬ 𝑦∅𝑧 | |
3 | 2 | intnan 951 | . . . . 5 ⊢ ¬ (𝑥 = 〈𝑧, 𝑦〉 ∧ 𝑦∅𝑧) |
4 | 3 | nex 1722 | . . . 4 ⊢ ¬ ∃𝑦(𝑥 = 〈𝑧, 𝑦〉 ∧ 𝑦∅𝑧) |
5 | 4 | nex 1722 | . . 3 ⊢ ¬ ∃𝑧∃𝑦(𝑥 = 〈𝑧, 𝑦〉 ∧ 𝑦∅𝑧) |
6 | df-cnv 5046 | . . . . 5 ⊢ ◡∅ = {〈𝑧, 𝑦〉 ∣ 𝑦∅𝑧} | |
7 | df-opab 4644 | . . . . 5 ⊢ {〈𝑧, 𝑦〉 ∣ 𝑦∅𝑧} = {𝑥 ∣ ∃𝑧∃𝑦(𝑥 = 〈𝑧, 𝑦〉 ∧ 𝑦∅𝑧)} | |
8 | 6, 7 | eqtri 2632 | . . . 4 ⊢ ◡∅ = {𝑥 ∣ ∃𝑧∃𝑦(𝑥 = 〈𝑧, 𝑦〉 ∧ 𝑦∅𝑧)} |
9 | 8 | abeq2i 2722 | . . 3 ⊢ (𝑥 ∈ ◡∅ ↔ ∃𝑧∃𝑦(𝑥 = 〈𝑧, 𝑦〉 ∧ 𝑦∅𝑧)) |
10 | 5, 9 | mtbir 312 | . 2 ⊢ ¬ 𝑥 ∈ ◡∅ |
11 | 1, 10 | mpgbir 1717 | 1 ⊢ ◡∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 ∅c0 3874 〈cop 4131 class class class wbr 4583 {copab 4642 ◡ccnv 5037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-nul 3875 df-br 4584 df-opab 4644 df-cnv 5046 |
This theorem is referenced by: xp0 5471 cnveq0 5509 co01 5567 funcnv0 5869 f10 6081 f1o00 6083 tpos0 7269 oduleval 16954 ust0 21833 nghmfval 22336 isnghm 22337 1pthonlem1 26119 mthmval 30726 resnonrel 36917 cononrel1 36919 cononrel2 36920 cnvrcl0 36951 0cnf 38762 mbf0 38849 1pthdlem1 41302 |
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