Theorem mainer 39260

Description: The Main Theorem of Equivalences: every equivalence relation implies equivalent comembers. (Contributed by Peter Mazsa, 26-Sep-2021.)

Assertion

Ref	Expression
mainer	⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴)

Proof of Theorem mainer

Step	Hyp	Ref	Expression
1		eqvrelqseqdisj2 39244	. . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ElDisj 𝐴)
2		eldisjim 39199	. . . 4 ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴)
3	1, 2	syl 17	. . 3 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → CoElEqvRel 𝐴)
4		n0eldmqseq 39046	. . . . 5 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴)
5	4	adantl 481	. . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ¬ ∅ ∈ 𝐴)
6		eldisjn0el 39221	. . . . 5 ⊢ ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
7	1, 6	syl 17	. . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
8	5, 7	mpbid 232	. . 3 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (∪ 𝐴 / ∼ 𝐴) = 𝐴)
9	3, 8	jca 511	. 2 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
10		dferALTV2 39065	. 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
11		dfcomember3 39071	. 2 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
12	9, 10, 11	3imtr4i 292	1 ⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∅c0 4274  ∪ cuni 4851  dom cdm 5622   / cqs 8633   ∼ ccoels 38496   EqvRel weqvrel 38512   CoElEqvRel wcoeleqvrel 38514   ErALTV werALTV 38521   CoMembEr wcomember 38525   ElDisj weldisj 38533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5517  df-eprel 5522  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ec 8636  df-qs 8640  df-coss 38813  df-coels 38814  df-refrel 38904  df-cnvrefrel 38919  df-symrel 38936  df-trrel 38970  df-eqvrel 38981  df-coeleqvrel 38983  df-dmqs 39035  df-erALTV 39061  df-comember 39063  df-funALTV 39079  df-disjALTV 39102  df-eldisj 39104

This theorem is referenced by:  partimcomember  39261  fences  39270