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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 4631 | . 2 ⊢ ¬ 𝑌∅𝑋 | |
2 | cnvnonrel 36913 | . . . 4 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ | |
3 | 2 | breqi 4589 | . . 3 ⊢ (𝑌◡(𝐴 ∖ ◡◡𝐴)𝑋 ↔ 𝑌∅𝑋) |
4 | brcnvg 5225 | . . . 4 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑈) → (𝑌◡(𝐴 ∖ ◡◡𝐴)𝑋 ↔ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌)) | |
5 | 4 | ancoms 468 | . . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑌◡(𝐴 ∖ ◡◡𝐴)𝑋 ↔ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌)) |
6 | 3, 5 | syl5rbbr 274 | . 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑋(𝐴 ∖ ◡◡𝐴)𝑌 ↔ 𝑌∅𝑋)) |
7 | 1, 6 | mtbiri 316 | 1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∖ cdif 3537 ∅c0 3874 class class class wbr 4583 ◡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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 |
This theorem is referenced by: (None) |
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