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Mirrors > Home > MPE Home > Th. List > Mathboxes > elnonrel | Structured version Visualization version GIF version |
Description: Only an ordered pair where not both entries are sets could be an element of the non-relation part of class. (Contributed by RP, 25-Oct-2020.) |
Ref | Expression |
---|---|
elnonrel | ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ ◡◡𝐴) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nonrel 43574 | . . 3 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V)) | |
2 | 1 | eleq2i 2831 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ ◡◡𝐴) ↔ 〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V))) |
3 | eldif 3973 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V)) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ 〈𝑋, 𝑌〉 ∈ (V × V))) | |
4 | opelxp 5725 | . . . . . 6 ⊢ (〈𝑋, 𝑌〉 ∈ (V × V) ↔ (𝑋 ∈ V ∧ 𝑌 ∈ V)) | |
5 | 4 | notbii 320 | . . . . 5 ⊢ (¬ 〈𝑋, 𝑌〉 ∈ (V × V) ↔ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
6 | 5 | anbi2i 623 | . . . 4 ⊢ ((〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ 〈𝑋, 𝑌〉 ∈ (V × V)) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
7 | opprc 4901 | . . . . . 6 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈𝑋, 𝑌〉 = ∅) | |
8 | 7 | eleq1d 2824 | . . . . 5 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (〈𝑋, 𝑌〉 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) |
9 | 8 | pm5.32ri 575 | . . . 4 ⊢ ((〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
10 | 6, 9 | bitri 275 | . . 3 ⊢ ((〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ 〈𝑋, 𝑌〉 ∈ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
11 | 3, 10 | bitri 275 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
12 | 2, 11 | bitri 275 | 1 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ ◡◡𝐴) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ∅c0 4339 〈cop 4637 × cxp 5687 ◡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-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 |
This theorem is referenced by: (None) |
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