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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 36909 | . . 3 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V)) | |
2 | 1 | eleq2i 2680 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ ◡◡𝐴) ↔ 〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V))) |
3 | eldif 3550 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V)) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ 〈𝑋, 𝑌〉 ∈ (V × V))) | |
4 | opelxp 5070 | . . . . . 6 ⊢ (〈𝑋, 𝑌〉 ∈ (V × V) ↔ (𝑋 ∈ V ∧ 𝑌 ∈ V)) | |
5 | 4 | notbii 309 | . . . . 5 ⊢ (¬ 〈𝑋, 𝑌〉 ∈ (V × V) ↔ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
6 | 5 | anbi2i 726 | . . . 4 ⊢ ((〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ 〈𝑋, 𝑌〉 ∈ (V × V)) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
7 | opprc 4362 | . . . . . 6 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈𝑋, 𝑌〉 = ∅) | |
8 | 7 | eleq1d 2672 | . . . . 5 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (〈𝑋, 𝑌〉 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) |
9 | 8 | pm5.32ri 668 | . . . 4 ⊢ ((〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
10 | 6, 9 | bitri 263 | . . 3 ⊢ ((〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ 〈𝑋, 𝑌〉 ∈ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
11 | 3, 10 | bitri 263 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
12 | 2, 11 | bitri 263 | 1 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ ◡◡𝐴) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 〈cop 4131 × cxp 5036 ◡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-rex 2902 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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