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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 39937 | . . 3 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V)) | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ ◡◡𝐴) ↔ 〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V))) |
3 | eldif 3945 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ (V × V)) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ ¬ 〈𝑋, 𝑌〉 ∈ (V × V))) | |
4 | opelxp 5585 | . . . . . 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 4819 | . . . . . 6 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈𝑋, 𝑌〉 = ∅) | |
8 | 7 | eleq1d 2897 | . . . . 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 2110 Vcvv 3494 ∖ cdif 3932 ∅c0 4290 〈cop 4566 × cxp 5547 ◡ccnv 5548 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 |
This theorem is referenced by: (None) |
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