Theorem cossssid3 37852

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.)

Assertion

Ref	Expression
cossssid3	⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))

Distinct variable group:   𝑢,𝑅,𝑥,𝑦

Proof of Theorem cossssid3

Step	Hyp	Ref	Expression
1		cossssid2 37851	. 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
2		19.23v 1937	. . . . 5 ⊢ (∀𝑢((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ (∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
3	2	albii 1813	. . . 4 ⊢ (∀𝑦∀𝑢((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
4		alcom 2148	. . . 4 ⊢ (∀𝑦∀𝑢((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑢∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
5	3, 4	bitr3i 277	. . 3 ⊢ (∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑢∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
6	5	albii 1813	. 2 ⊢ (∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑢∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
7		alcom 2148	. 2 ⊢ (∀𝑥∀𝑢∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
8	1, 6, 7	3bitri 297	1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531  ∃wex 1773   ⊆ wss 3943   class class class wbr 5141   I cid 5566   ≀ ccoss 37556

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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204  df-id 5567  df-coss 37794

This theorem is referenced by:  cossssid4  37853  cosscnvssid3  37859  cosselcnvrefrels3  37922  dffunALTV3  38072