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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cossssid4 | 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, 31-Aug-2021.) |
Ref | Expression |
---|---|
cossssid4 | ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossssid3 37795 | . 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) | |
2 | breq2 5142 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑢𝑅𝑥 ↔ 𝑢𝑅𝑦)) | |
3 | 2 | mo4 2552 | . . 3 ⊢ (∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) |
4 | 3 | albii 1813 | . 2 ⊢ (∀𝑢∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) |
5 | 1, 4 | bitr4i 278 | 1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 ∃*wmo 2524 ⊆ wss 3940 class class class wbr 5138 I cid 5563 ≀ ccoss 37499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-id 5564 df-coss 37737 |
This theorem is referenced by: cossssid5 37797 cosscnvssid4 37803 cosselcnvrefrels4 37866 dffunALTV4 38016 |
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