Theorem djuen 7416

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 6910	. . . . . . . 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 7287	. . . . . 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 6953	. . . . 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 7287	. . . . . 6 ⊢ (𝐵 ∈ V → (inl “ 𝐵) ≈ 𝐵)
10	2, 9	simpl2im 386	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inl “ 𝐵) ≈ 𝐵)
11	10	ensymd 6952	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐵 ≈ (inl “ 𝐵))
12		entr 6953	. . . 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 6910	. . . . . . . 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 7288	. . . . . 6 ⊢ (𝐶 ∈ V → (inr “ 𝐶) ≈ 𝐶)
18	16, 17	syl 14	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐶)
19		entr 6953	. . . . 5 ⊢ (((inr “ 𝐶) ≈ 𝐶 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐷)
20	18, 19	sylancom 420	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐷)
21		eninr 7288	. . . . . 6 ⊢ (𝐷 ∈ V → (inr “ 𝐷) ≈ 𝐷)
22	15, 21	simpl2im 386	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐷) ≈ 𝐷)
23	22	ensymd 6952	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐷 ≈ (inr “ 𝐷))
24		entr 6953	. . . 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 7254	. . . 4 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅
27	26	a1i 9	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅)
28		djuin 7254	. . . 4 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅
29	28	a1i 9	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅)
30		unen 6986	. . 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 1272	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐴) ∪ (inr “ 𝐶)) ≈ ((inl “ 𝐵) ∪ (inr “ 𝐷)))
32		djuun 7257	. 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐶)) = (𝐴 ⊔ 𝐶)
33		djuun 7257	. 2 ⊢ ((inl “ 𝐵) ∪ (inr “ 𝐷)) = (𝐵 ⊔ 𝐷)
34	31, 32, 33	3brtr3g 4119	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2800   ∪ cun 3196   ∩ cin 3197  ∅c0 3492   class class class wbr 4086   “ cima 4726   ≈ cen 6902   ⊔ cdju 7227  inlcinl 7235  inrcinr 7236

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1st 6298  df-2nd 6299  df-1o 6577  df-er 6697  df-en 6905  df-dju 7228  df-inl 7237  df-inr 7238

This theorem is referenced by:  djuenun  7417  exmidunben  13037  enctlem  13043