Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opeldifid | Structured version Visualization version GIF version |
Description: Ordered pair elementhood outside of the diagonal. (Contributed by Thierry Arnoux, 1-Jan-2020.) |
Ref | Expression |
---|---|
opeldifid | ⊢ (Rel 𝐴 → (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ I ) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldif 5161 | . . . 4 ⊢ (Rel 𝐴 → Rel (𝐴 ∖ I )) | |
2 | brrelex2 5081 | . . . 4 ⊢ ((Rel (𝐴 ∖ I ) ∧ 𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V) | |
3 | 1, 2 | sylan 487 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V) |
4 | brrelex2 5081 | . . . 4 ⊢ ((Rel 𝐴 ∧ 𝑋𝐴𝑌) → 𝑌 ∈ V) | |
5 | 4 | adantrr 749 | . . 3 ⊢ ((Rel 𝐴 ∧ (𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ V) |
6 | brdif 4635 | . . . 4 ⊢ (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌)) | |
7 | ideqg 5195 | . . . . . 6 ⊢ (𝑌 ∈ V → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
8 | 7 | necon3bbid 2819 | . . . . 5 ⊢ (𝑌 ∈ V → (¬ 𝑋 I 𝑌 ↔ 𝑋 ≠ 𝑌)) |
9 | 8 | anbi2d 736 | . . . 4 ⊢ (𝑌 ∈ V → ((𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌) ↔ (𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌))) |
10 | 6, 9 | syl5bb 271 | . . 3 ⊢ (𝑌 ∈ V → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌))) |
11 | 3, 5, 10 | pm5.21nd 939 | . 2 ⊢ (Rel 𝐴 → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌))) |
12 | df-br 4584 | . 2 ⊢ (𝑋(𝐴 ∖ I )𝑌 ↔ 〈𝑋, 𝑌〉 ∈ (𝐴 ∖ I )) | |
13 | df-br 4584 | . . 3 ⊢ (𝑋𝐴𝑌 ↔ 〈𝑋, 𝑌〉 ∈ 𝐴) | |
14 | 13 | anbi1i 727 | . 2 ⊢ ((𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) |
15 | 11, 12, 14 | 3bitr3g 301 | 1 ⊢ (Rel 𝐴 → (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ I ) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∖ cdif 3537 〈cop 4131 class class class wbr 4583 I cid 4948 Rel wrel 5043 |
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-ne 2782 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-id 4953 df-xp 5044 df-rel 5045 |
This theorem is referenced by: qtophaus 29231 |
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