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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 5687 | . . . 4 ⊢ (Rel 𝐴 → Rel (𝐴 ∖ I )) | |
2 | brrelex2 5605 | . . . 4 ⊢ ((Rel (𝐴 ∖ I ) ∧ 𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V) | |
3 | 1, 2 | sylan 582 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝑋(𝐴 ∖ I )𝑌) → 𝑌 ∈ V) |
4 | brrelex2 5605 | . . . 4 ⊢ ((Rel 𝐴 ∧ 𝑋𝐴𝑌) → 𝑌 ∈ V) | |
5 | 4 | adantrr 715 | . . 3 ⊢ ((Rel 𝐴 ∧ (𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ V) |
6 | brdif 5118 | . . . 4 ⊢ (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌)) | |
7 | ideqg 5721 | . . . . . 6 ⊢ (𝑌 ∈ V → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
8 | 7 | necon3bbid 3053 | . . . . 5 ⊢ (𝑌 ∈ V → (¬ 𝑋 I 𝑌 ↔ 𝑋 ≠ 𝑌)) |
9 | 8 | anbi2d 630 | . . . 4 ⊢ (𝑌 ∈ V → ((𝑋𝐴𝑌 ∧ ¬ 𝑋 I 𝑌) ↔ (𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌))) |
10 | 6, 9 | syl5bb 285 | . . 3 ⊢ (𝑌 ∈ V → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌))) |
11 | 3, 5, 10 | pm5.21nd 800 | . 2 ⊢ (Rel 𝐴 → (𝑋(𝐴 ∖ I )𝑌 ↔ (𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌))) |
12 | df-br 5066 | . 2 ⊢ (𝑋(𝐴 ∖ I )𝑌 ↔ 〈𝑋, 𝑌〉 ∈ (𝐴 ∖ I )) | |
13 | df-br 5066 | . . 3 ⊢ (𝑋𝐴𝑌 ↔ 〈𝑋, 𝑌〉 ∈ 𝐴) | |
14 | 13 | anbi1i 625 | . 2 ⊢ ((𝑋𝐴𝑌 ∧ 𝑋 ≠ 𝑌) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) |
15 | 11, 12, 14 | 3bitr3g 315 | 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 208 ∧ wa 398 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∖ cdif 3932 〈cop 4572 class class class wbr 5065 I cid 5458 Rel wrel 5559 |
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 5202 ax-nul 5209 ax-pr 5329 |
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-ne 3017 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 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 |
This theorem is referenced by: qtophaus 31100 |
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