Theorem disjim 39796

Description: The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 39916, cf. eldisjim 39799. (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)

Assertion

Ref	Expression
disjim	⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)

Proof of Theorem disjim

Dummy variables 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdisjALTV4 39713	. . . 4 ⊢ ( Disj 𝑅 ↔ (∀𝑦∃*𝑢 𝑢𝑅𝑦 ∧ Rel 𝑅))
2	1	simplbi 502	. . 3 ⊢ ( Disj 𝑅 → ∀𝑦∃*𝑢 𝑢𝑅𝑦)
3		trcoss 39484	. . 3 ⊢ (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
4	2, 3	syl 18	. 2 ⊢ ( Disj 𝑅 → ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
5		eqvrelcoss3 39614	. 2 ⊢ ( EqvRel ≀ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
6	4, 5	sylibr 237	1 ⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃*wmo 2563   class class class wbr 5103  Rel wrel 5656   ≀ ccoss 39095   EqvRel weqvrel 39112   Disj wdisjALTV 39131

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-coss 39413  df-refrel 39504  df-cnvrefrel 39519  df-symrel 39536  df-trrel 39570  df-eqvrel 39581  df-disjALTV 39702

This theorem is used by:  disjimi  39797  detlem  39798  eldisjim  39799  eldisjim2  39800  partim2  39822  disjimeldisjdmqs  39845