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Mirrors > Home > MPE Home > Th. List > Mathboxes > cossssid3 | Structured version Visualization version GIF version |
Description: Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 10-Mar-2019.) |
Ref | Expression |
---|---|
cossssid3 | ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossssid2 35588 | . 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) | |
2 | 19.23v 1934 | . . . . 5 ⊢ (∀𝑢((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ (∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) | |
3 | 2 | albii 1811 | . . . 4 ⊢ (∀𝑦∀𝑢((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) |
4 | alcom 2153 | . . . 4 ⊢ (∀𝑦∀𝑢((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑢∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) | |
5 | 3, 4 | bitr3i 278 | . . 3 ⊢ (∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑢∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) |
6 | 5 | albii 1811 | . 2 ⊢ (∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑢∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) |
7 | alcom 2153 | . 2 ⊢ (∀𝑥∀𝑢∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) | |
8 | 1, 6, 7 | 3bitri 298 | 1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 ∃wex 1771 ⊆ wss 3933 class class class wbr 5057 I cid 5452 ≀ ccoss 35334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-id 5453 df-coss 35539 |
This theorem is referenced by: cossssid4 35590 cosscnvssid3 35596 cosselcnvrefrels3 35655 dffunALTV3 35802 |
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