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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 5238 | . . . 4 ⊢ (Rel 𝐴 → Rel (𝐴 ∖ I )) | |
2 | brrelex2 5157 | . . . 4 ⊢ ((Rel (𝐴 ∖ I ) ∧ 𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V) | |
3 | 1, 2 | sylan 488 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V) |
4 | brrelex2 5157 | . . . 4 ⊢ ((Rel 𝐴 ∧ 𝑋𝐴𝑌) → 𝑌 ∈ V) | |
5 | 4 | adantrr 753 | . . 3 ⊢ ((Rel 𝐴 ∧ (𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ V) |
6 | brdif 4705 | . . . 4 ⊢ (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌)) | |
7 | ideqg 5273 | . . . . . 6 ⊢ (𝑌 ∈ V → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
8 | 7 | necon3bbid 2831 | . . . . 5 ⊢ (𝑌 ∈ V → (¬ 𝑋 I 𝑌 ↔ 𝑋 ≠ 𝑌)) |
9 | 8 | anbi2d 740 | . . . 4 ⊢ (𝑌 ∈ V → ((𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌) ↔ (𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌))) |
10 | 6, 9 | syl5bb 272 | . . 3 ⊢ (𝑌 ∈ V → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌))) |
11 | 3, 5, 10 | pm5.21nd 941 | . 2 ⊢ (Rel 𝐴 → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌))) |
12 | df-br 4654 | . 2 ⊢ (𝑋(𝐴 ∖ I )𝑌 ↔ 〈𝑋, 𝑌〉 ∈ (𝐴 ∖ I )) | |
13 | df-br 4654 | . . 3 ⊢ (𝑋𝐴𝑌 ↔ 〈𝑋, 𝑌〉 ∈ 𝐴) | |
14 | 13 | anbi1i 731 | . 2 ⊢ ((𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) |
15 | 11, 12, 14 | 3bitr3g 302 | 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 196 ∧ wa 384 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∖ cdif 3571 〈cop 4183 class class class wbr 4653 I cid 5023 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 |
This theorem is referenced by: qtophaus 29903 |
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