Theorem djuen 7349

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 6851	. . . . . . . 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 7220	. . . . . 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 6894	. . . . 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 7220	. . . . . 6 ⊢ (𝐵 ∈ V → (inl “ 𝐵) ≈ 𝐵)
10	2, 9	simpl2im 386	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inl “ 𝐵) ≈ 𝐵)
11	10	ensymd 6893	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐵 ≈ (inl “ 𝐵))
12		entr 6894	. . . 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 6851	. . . . . . . 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 7221	. . . . . 6 ⊢ (𝐶 ∈ V → (inr “ 𝐶) ≈ 𝐶)
18	16, 17	syl 14	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐶)
19		entr 6894	. . . . 5 ⊢ (((inr “ 𝐶) ≈ 𝐶 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐷)
20	18, 19	sylancom 420	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐷)
21		eninr 7221	. . . . . 6 ⊢ (𝐷 ∈ V → (inr “ 𝐷) ≈ 𝐷)
22	15, 21	simpl2im 386	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐷) ≈ 𝐷)
23	22	ensymd 6893	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐷 ≈ (inr “ 𝐷))
24		entr 6894	. . . 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 7187	. . . 4 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅
27	26	a1i 9	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅)
28		djuin 7187	. . . 4 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅
29	28	a1i 9	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅)
30		unen 6927	. . 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 1251	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐴) ∪ (inr “ 𝐶)) ≈ ((inl “ 𝐵) ∪ (inr “ 𝐷)))
32		djuun 7190	. 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐶)) = (𝐴 ⊔ 𝐶)
33		djuun 7190	. 2 ⊢ ((inl “ 𝐵) ∪ (inr “ 𝐷)) = (𝐵 ⊔ 𝐷)
34	31, 32, 33	3brtr3g 4087	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  Vcvv 2773   ∪ cun 3168   ∩ cin 3169  ∅c0 3464   class class class wbr 4054   “ cima 4691   ≈ cen 6843   ⊔ cdju 7160  inlcinl 7168  inrcinr 7169

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-1st 6244  df-2nd 6245  df-1o 6520  df-er 6638  df-en 6846  df-dju 7161  df-inl 7170  df-inr 7171

This theorem is referenced by:  djuenun  7350  exmidunben  12882  enctlem  12888