![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cossssid2 | 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 |
---|---|
cossssid2 | ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-id 5597 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
2 | 1 | sseq2i 4032 | . 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦}) |
3 | df-coss 38315 | . . 3 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
4 | 3 | sseq1i 4031 | . 2 ⊢ ( ≀ 𝑅 ⊆ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} ↔ {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} ⊆ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦}) |
5 | ssopab2bw 5570 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} ⊆ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} ↔ ∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) | |
6 | 2, 4, 5 | 3bitri 297 | 1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 ∃wex 1777 ⊆ wss 3970 class class class wbr 5169 {copab 5231 I cid 5596 ≀ ccoss 38083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-opab 5232 df-id 5597 df-coss 38315 |
This theorem is referenced by: cossssid3 38373 |
Copyright terms: Public domain | W3C validator |