Theorem djuen 7278

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 6805	. . . . . . . 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 7163	. . . . . 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 6843	. . . . 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 7163	. . . . . 6 ⊢ (𝐵 ∈ V → (inl “ 𝐵) ≈ 𝐵)
10	2, 9	simpl2im 386	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inl “ 𝐵) ≈ 𝐵)
11	10	ensymd 6842	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐵 ≈ (inl “ 𝐵))
12		entr 6843	. . . 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 6805	. . . . . . . 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 7164	. . . . . 6 ⊢ (𝐶 ∈ V → (inr “ 𝐶) ≈ 𝐶)
18	16, 17	syl 14	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐶)
19		entr 6843	. . . . 5 ⊢ (((inr “ 𝐶) ≈ 𝐶 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐷)
20	18, 19	sylancom 420	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐷)
21		eninr 7164	. . . . . 6 ⊢ (𝐷 ∈ V → (inr “ 𝐷) ≈ 𝐷)
22	15, 21	simpl2im 386	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐷) ≈ 𝐷)
23	22	ensymd 6842	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐷 ≈ (inr “ 𝐷))
24		entr 6843	. . . 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 7130	. . . 4 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅
27	26	a1i 9	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅)
28		djuin 7130	. . . 4 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅
29	28	a1i 9	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅)
30		unen 6875	. . 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 7133	. 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐶)) = (𝐴 ⊔ 𝐶)
33		djuun 7133	. 2 ⊢ ((inl “ 𝐵) ∪ (inr “ 𝐷)) = (𝐵 ⊔ 𝐷)
34	31, 32, 33	3brtr3g 4066	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∪ cun 3155   ∩ cin 3156  ∅c0 3450   class class class wbr 4033   “ cima 4666   ≈ cen 6797   ⊔ cdju 7103  inlcinl 7111  inrcinr 7112

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-er 6592  df-en 6800  df-dju 7104  df-inl 7113  df-inr 7114

This theorem is referenced by:  djuenun  7279  exmidunben  12643  enctlem  12649