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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 3838 | . . . 4 ⊢ (𝑅 ∩ I ) = ( I ∩ 𝑅) | |
2 | 1 | eqeq1i 2656 | . . 3 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅) |
3 | disj2 4057 | . . 3 ⊢ (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅)) | |
4 | reli 5282 | . . . 4 ⊢ Rel I | |
5 | ssrel 5241 | . . . 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 1991 | . . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
9 | vex 3234 | . . . . . 6 ⊢ 𝑦 ∈ V | |
10 | 9 | ideq 5307 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
11 | df-br 4686 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
12 | 8, 10, 11 | 3bitr2i 288 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 〈𝑥, 𝑦〉 ∈ I ) |
13 | opex 4962 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
14 | 13 | biantrur 526 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ 𝑅 ↔ (〈𝑥, 𝑦〉 ∈ V ∧ ¬ 〈𝑥, 𝑦〉 ∈ 𝑅)) |
15 | eldif 3617 | . . . . . 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 4686 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
18 | 16, 17 | xchnxbir 322 | . . . 4 ⊢ (¬ 𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅)) |
19 | 12, 18 | imbi12i 339 | . . 3 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ (〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅))) |
20 | 19 | 2albii 1788 | . 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅))) |
21 | breq2 4689 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑥)) | |
22 | 21 | notbid 307 | . . . 4 ⊢ (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥)) |
23 | 22 | equsalvw 1977 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ¬ 𝑥𝑅𝑥) |
24 | 23 | albii 1787 | . 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 383 ∀wal 1521 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∖ cdif 3604 ∩ cin 3606 ⊆ wss 3607 ∅c0 3948 〈cop 4216 class class class wbr 4685 I cid 5052 Rel wrel 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 |
This theorem is referenced by: hartogslem1 8488 hausdiag 21496 |
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