Theorem djuen 7322

Description: Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)

Assertion

Ref	Expression
djuen	⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷))

Proof of Theorem djuen

Step	Hyp	Ref	Expression
1		encv 6832	. . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2	1	adantr 276	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	simpld 112	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐴 ∈ V)
4		eninl 7198	. . . . . 6 ⊢ (𝐴 ∈ V → (inl “ 𝐴) ≈ 𝐴)
5	3, 4	syl 14	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inl “ 𝐴) ≈ 𝐴)
6		simpl 109	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐴 ≈ 𝐵)
7		entr 6875	. . . . 5 ⊢ (((inl “ 𝐴) ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → (inl “ 𝐴) ≈ 𝐵)
8	5, 6, 7	syl2anc 411	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inl “ 𝐴) ≈ 𝐵)
9		eninl 7198	. . . . . 6 ⊢ (𝐵 ∈ V → (inl “ 𝐵) ≈ 𝐵)
10	2, 9	simpl2im 386	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inl “ 𝐵) ≈ 𝐵)
11	10	ensymd 6874	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐵 ≈ (inl “ 𝐵))
12		entr 6875	. . . 4 ⊢ (((inl “ 𝐴) ≈ 𝐵 ∧ 𝐵 ≈ (inl “ 𝐵)) → (inl “ 𝐴) ≈ (inl “ 𝐵))
13	8, 11, 12	syl2anc 411	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inl “ 𝐴) ≈ (inl “ 𝐵))
14		encv 6832	. . . . . . . 8 ⊢ (𝐶 ≈ 𝐷 → (𝐶 ∈ V ∧ 𝐷 ∈ V))
15	14	adantl 277	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐶 ∈ V ∧ 𝐷 ∈ V))
16	15	simpld 112	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐶 ∈ V)
17		eninr 7199	. . . . . 6 ⊢ (𝐶 ∈ V → (inr “ 𝐶) ≈ 𝐶)
18	16, 17	syl 14	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐶)
19		entr 6875	. . . . 5 ⊢ (((inr “ 𝐶) ≈ 𝐶 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐷)
20	18, 19	sylancom 420	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐷)
21		eninr 7199	. . . . . 6 ⊢ (𝐷 ∈ V → (inr “ 𝐷) ≈ 𝐷)
22	15, 21	simpl2im 386	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐷) ≈ 𝐷)
23	22	ensymd 6874	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐷 ≈ (inr “ 𝐷))
24		entr 6875	. . . 4 ⊢ (((inr “ 𝐶) ≈ 𝐷 ∧ 𝐷 ≈ (inr “ 𝐷)) → (inr “ 𝐶) ≈ (inr “ 𝐷))
25	20, 23, 24	syl2anc 411	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ (inr “ 𝐷))
26		djuin 7165	. . . 4 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅
27	26	a1i 9	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅)
28		djuin 7165	. . . 4 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅
29	28	a1i 9	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅)
30		unen 6907	. . 3 ⊢ ((((inl “ 𝐴) ≈ (inl “ 𝐵) ∧ (inr “ 𝐶) ≈ (inr “ 𝐷)) ∧ (((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅ ∧ ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅)) → ((inl “ 𝐴) ∪ (inr “ 𝐶)) ≈ ((inl “ 𝐵) ∪ (inr “ 𝐷)))
31	13, 25, 27, 29, 30	syl22anc 1250	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐴) ∪ (inr “ 𝐶)) ≈ ((inl “ 𝐵) ∪ (inr “ 𝐷)))
32		djuun 7168	. 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐶)) = (𝐴 ⊔ 𝐶)
33		djuun 7168	. 2 ⊢ ((inl “ 𝐵) ∪ (inr “ 𝐷)) = (𝐵 ⊔ 𝐷)
34	31, 32, 33	3brtr3g 4076	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372   ∈ wcel 2175  Vcvv 2771   ∪ cun 3163   ∩ cin 3164  ∅c0 3459   class class class wbr 4043   “ cima 4677   ≈ cen 6824   ⊔ cdju 7138  inlcinl 7146  inrcinr 7147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-1st 6225  df-2nd 6226  df-1o 6501  df-er 6619  df-en 6827  df-dju 7139  df-inl 7148  df-inr 7149

This theorem is referenced by:  djuenun  7323  exmidunben  12768  enctlem  12774