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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 4178 | . . . 4 ⊢ (𝑅 ∩ I ) = ( I ∩ 𝑅) | |
2 | 1 | eqeq1i 2826 | . . 3 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅) |
3 | disj2 4407 | . . 3 ⊢ (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅)) | |
4 | reli 5698 | . . . 4 ⊢ Rel I | |
5 | ssrel 5657 | . . . 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 299 | . 2 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅))) |
8 | equcom 2025 | . . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
9 | vex 3497 | . . . . . 6 ⊢ 𝑦 ∈ V | |
10 | 9 | ideq 5723 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
11 | df-br 5067 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
12 | 8, 10, 11 | 3bitr2i 301 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 〈𝑥, 𝑦〉 ∈ I ) |
13 | opex 5356 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
14 | 13 | biantrur 533 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ 𝑅 ↔ (〈𝑥, 𝑦〉 ∈ V ∧ ¬ 〈𝑥, 𝑦〉 ∈ 𝑅)) |
15 | eldif 3946 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅) ↔ (〈𝑥, 𝑦〉 ∈ V ∧ ¬ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
16 | 14, 15 | bitr4i 280 | . . . . 5 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅)) |
17 | df-br 5067 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
18 | 16, 17 | xchnxbir 335 | . . . 4 ⊢ (¬ 𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅)) |
19 | 12, 18 | imbi12i 353 | . . 3 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ (〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅))) |
20 | 19 | 2albii 1821 | . 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ (V ∖ 𝑅))) |
21 | breq2 5070 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑥)) | |
22 | 21 | notbid 320 | . . . 4 ⊢ (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥)) |
23 | 22 | equsalvw 2010 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ¬ 𝑥𝑅𝑥) |
24 | 23 | albii 1820 | . 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥 ¬ 𝑥𝑅𝑥) |
25 | 7, 20, 24 | 3bitr2i 301 | 1 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 〈cop 4573 class class class wbr 5066 I cid 5459 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 |
This theorem is referenced by: hartogslem1 9006 hausdiag 22253 |
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