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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 4128 | . . . 4 ⊢ (𝑅 ∩ I ) = ( I ∩ 𝑅) | |
2 | 1 | eqeq1i 2803 | . . 3 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅) |
3 | disj2 4365 | . . 3 ⊢ (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅)) | |
4 | reli 5662 | . . . 4 ⊢ Rel I | |
5 | ssrel 5621 | . . . 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 300 | . 2 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅))) |
8 | equcom 2025 | . . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
9 | vex 3444 | . . . . . 6 ⊢ 𝑦 ∈ V | |
10 | 9 | ideq 5687 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
11 | df-br 5031 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
12 | 8, 10, 11 | 3bitr2i 302 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 〈𝑥, 𝑦〉 ∈ I ) |
13 | opex 5321 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
14 | 13 | biantrur 534 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ 𝑅 ↔ (〈𝑥, 𝑦〉 ∈ V ∧ ¬ 〈𝑥, 𝑦〉 ∈ 𝑅)) |
15 | eldif 3891 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅) ↔ (〈𝑥, 𝑦〉 ∈ V ∧ ¬ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
16 | 14, 15 | bitr4i 281 | . . . . 5 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅)) |
17 | df-br 5031 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
18 | 16, 17 | xchnxbir 336 | . . . 4 ⊢ (¬ 𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅)) |
19 | 12, 18 | imbi12i 354 | . . 3 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ (〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅))) |
20 | 19 | 2albii 1822 | . 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅))) |
21 | breq2 5034 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑥)) | |
22 | 21 | notbid 321 | . . . 4 ⊢ (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥)) |
23 | 22 | equsalvw 2010 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ¬ 𝑥𝑅𝑥) |
24 | 23 | albii 1821 | . 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥 ¬ 𝑥𝑅𝑥) |
25 | 7, 20, 24 | 3bitr2i 302 | 1 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 〈cop 4531 class class class wbr 5030 I cid 5424 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 |
This theorem is referenced by: hartogslem1 8990 hausdiag 22250 |
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