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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 43597 | . . 3 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V)) | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ ◡◡𝐴) ↔ 〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V))) |
| 3 | eldif 3961 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V)) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ 〈𝑋, 𝑌〉 ∈ (V × V))) | |
| 4 | opelxp 5721 | . . . . . 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 4896 | . . . . . 6 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈𝑋, 𝑌〉 = ∅) | |
| 8 | 7 | eleq1d 2826 | . . . . 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 2108 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 〈cop 4632 × cxp 5683 ◡ccnv 5684 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: (None) |
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