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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 39951 | . . 3 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V)) | |
2 | 1 | eleq2i 2906 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ ◡◡𝐴) ↔ 〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V))) |
3 | eldif 3948 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V)) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ 〈𝑋, 𝑌〉 ∈ (V × V))) | |
4 | opelxp 5593 | . . . . . 6 ⊢ (〈𝑋, 𝑌〉 ∈ (V × V) ↔ (𝑋 ∈ V ∧ 𝑌 ∈ V)) | |
5 | 4 | notbii 322 | . . . . 5 ⊢ (¬ 〈𝑋, 𝑌〉 ∈ (V × V) ↔ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
6 | 5 | anbi2i 624 | . . . 4 ⊢ ((〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ 〈𝑋, 𝑌〉 ∈ (V × V)) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
7 | opprc 4828 | . . . . . 6 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈𝑋, 𝑌〉 = ∅) | |
8 | 7 | eleq1d 2899 | . . . . 5 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (〈𝑋, 𝑌〉 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) |
9 | 8 | pm5.32ri 578 | . . . 4 ⊢ ((〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
10 | 6, 9 | bitri 277 | . . 3 ⊢ ((〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ 〈𝑋, 𝑌〉 ∈ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
11 | 3, 10 | bitri 277 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
12 | 2, 11 | bitri 277 | 1 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ ◡◡𝐴) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 ∅c0 4293 〈cop 4575 × cxp 5555 ◡ccnv 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 |
This theorem is referenced by: (None) |
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