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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cosscnvex | Structured version Visualization version GIF version |
Description: If 𝐴 is a set then the class of cosets by the converse of 𝐴 is a set. (Contributed by Peter Mazsa, 18-Oct-2019.) |
Ref | Expression |
---|---|
cosscnvex | ⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvexg 7902 | . 2 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) | |
2 | cossex 37195 | . 2 ⊢ (◡𝐴 ∈ V → ≀ ◡𝐴 ∈ V) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3475 ◡ccnv 5671 ≀ ccoss 36949 |
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-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 |
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-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-coss 37187 |
This theorem is referenced by: eldisjsdisj 37503 |
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