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Mirrors > Home > MPE Home > Th. List > intirr | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
intirr | ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3805 | . . . 4 ⊢ (𝑅 ∩ I ) = ( I ∩ 𝑅) | |
2 | 1 | eqeq1i 2627 | . . 3 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅) |
3 | disj2 4024 | . . 3 ⊢ (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅)) | |
4 | reli 5249 | . . . 4 ⊢ Rel I | |
5 | ssrel 5207 | . . . 4 ⊢ (Rel I → ( I ⊆ (V ∖ 𝑅) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅)))) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ( I ⊆ (V ∖ 𝑅) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅))) |
7 | 2, 3, 6 | 3bitri 286 | . 2 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅))) |
8 | equcom 1945 | . . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
9 | vex 3203 | . . . . . 6 ⊢ 𝑦 ∈ V | |
10 | 9 | ideq 5274 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
11 | df-br 4654 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
12 | 8, 10, 11 | 3bitr2i 288 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 〈𝑥, 𝑦〉 ∈ I ) |
13 | opex 4932 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
14 | 13 | biantrur 527 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ 𝑅 ↔ (〈𝑥, 𝑦〉 ∈ V ∧ ¬ 〈𝑥, 𝑦〉 ∈ 𝑅)) |
15 | eldif 3584 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅) ↔ (〈𝑥, 𝑦〉 ∈ V ∧ ¬ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
16 | 14, 15 | bitr4i 267 | . . . . 5 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅)) |
17 | df-br 4654 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
18 | 16, 17 | xchnxbir 323 | . . . 4 ⊢ (¬ 𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅)) |
19 | 12, 18 | imbi12i 340 | . . 3 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ (〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅))) |
20 | 19 | 2albii 1748 | . 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅))) |
21 | breq2 4657 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑥)) | |
22 | 21 | notbid 308 | . . . 4 ⊢ (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥)) |
23 | 22 | equsalvw 1931 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ¬ 𝑥𝑅𝑥) |
24 | 23 | albii 1747 | . 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥 ¬ 𝑥𝑅𝑥) |
25 | 7, 20, 24 | 3bitr2i 288 | 1 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 〈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-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: hartogslem1 8447 hausdiag 21448 |
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