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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 5302, ax-nul 5312, ax-pr 5438. (Revised by KP, 25-Oct-2021.) |
Ref | Expression |
---|---|
cnv0 | ⊢ ◡∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | br0 5197 | . . . . . 6 ⊢ ¬ 𝑦∅𝑧 | |
2 | 1 | intnan 486 | . . . . 5 ⊢ ¬ (𝑥 = 〈𝑧, 𝑦〉 ∧ 𝑦∅𝑧) |
3 | 2 | nex 1797 | . . . 4 ⊢ ¬ ∃𝑦(𝑥 = 〈𝑧, 𝑦〉 ∧ 𝑦∅𝑧) |
4 | 3 | nex 1797 | . . 3 ⊢ ¬ ∃𝑧∃𝑦(𝑥 = 〈𝑧, 𝑦〉 ∧ 𝑦∅𝑧) |
5 | df-cnv 5697 | . . . . 5 ⊢ ◡∅ = {〈𝑧, 𝑦〉 ∣ 𝑦∅𝑧} | |
6 | df-opab 5211 | . . . . 5 ⊢ {〈𝑧, 𝑦〉 ∣ 𝑦∅𝑧} = {𝑥 ∣ ∃𝑧∃𝑦(𝑥 = 〈𝑧, 𝑦〉 ∧ 𝑦∅𝑧)} | |
7 | 5, 6 | eqtri 2763 | . . . 4 ⊢ ◡∅ = {𝑥 ∣ ∃𝑧∃𝑦(𝑥 = 〈𝑧, 𝑦〉 ∧ 𝑦∅𝑧)} |
8 | 7 | eqabri 2883 | . . 3 ⊢ (𝑥 ∈ ◡∅ ↔ ∃𝑧∃𝑦(𝑥 = 〈𝑧, 𝑦〉 ∧ 𝑦∅𝑧)) |
9 | 4, 8 | mtbir 323 | . 2 ⊢ ¬ 𝑥 ∈ ◡∅ |
10 | 9 | nel0 4360 | 1 ⊢ ◡∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 ∅c0 4339 〈cop 4637 class class class wbr 5148 {copab 5210 ◡ccnv 5688 |
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-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-dif 3966 df-nul 4340 df-br 5149 df-opab 5211 df-cnv 5697 |
This theorem is referenced by: xp0 6180 cnveq0 6219 co01 6283 funcnv0 6634 f1o00 6884 tpos0 8280 cnvfi 9215 oduleval 18346 ust0 24244 nghmfval 24759 isnghm 24760 1pthdlem1 30164 mptiffisupp 32708 tocycf 33120 tocyc01 33121 mthmval 35560 resnonrel 43582 cononrel1 43584 cononrel2 43585 cnvrcl0 43615 0cnf 45833 |
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