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Mirrors > Home > MPE Home > Th. List > Mathboxes > brnonrel | Structured version Visualization version GIF version |
Description: A non-relation cannot relate any two classes. (Contributed by RP, 23-Oct-2020.) |
Ref | Expression |
---|---|
brnonrel | ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | br0 5106 | . 2 ⊢ ¬ 𝑌∅𝑋 | |
2 | cnvnonrel 39826 | . . . 4 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ | |
3 | 2 | breqi 5063 | . . 3 ⊢ (𝑌◡(𝐴 ∖ ◡◡𝐴)𝑋 ↔ 𝑌∅𝑋) |
4 | brcnvg 5743 | . . . 4 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑈) → (𝑌◡(𝐴 ∖ ◡◡𝐴)𝑋 ↔ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌)) | |
5 | 4 | ancoms 459 | . . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑌◡(𝐴 ∖ ◡◡𝐴)𝑋 ↔ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌)) |
6 | 3, 5 | syl5rbbr 287 | . 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑋(𝐴 ∖ ◡◡𝐴)𝑌 ↔ 𝑌∅𝑋)) |
7 | 1, 6 | mtbiri 328 | 1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ∖ cdif 3930 ∅c0 4288 class class class wbr 5057 ◡ccnv 5547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 |
This theorem is referenced by: (None) |
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