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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 5340 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
2 | 1 | sseq2i 3912 | . 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦}) |
3 | df-coss 35140 | . . 3 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
4 | 3 | sseq1i 3911 | . 2 ⊢ ( ≀ 𝑅 ⊆ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} ↔ {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} ⊆ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦}) |
5 | ssopab2b 5316 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} ⊆ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} ↔ ∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) | |
6 | 2, 4, 5 | 3bitri 298 | 1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1518 ∃wex 1759 ⊆ wss 3854 class class class wbr 4956 {copab 5018 I cid 5339 ≀ ccoss 34933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pr 5214 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ral 3108 df-rab 3112 df-v 3434 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-opab 5019 df-id 5340 df-coss 35140 |
This theorem is referenced by: cossssid3 35190 |
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