Theorem unen 6229

Description: Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
unen	⊢ (((A ≈ B ∧ 𝐶 ≈ 𝐷) ∧ ((A ∩ 𝐶) = ∅ ∧ (B ∩ 𝐷) = ∅)) → (A ∪ 𝐶) ≈ (B ∪ 𝐷))

Proof of Theorem unen

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6164	. . 3 ⊢ (A ≈ B ↔ ∃x x:A–1-1-onto→B)
2		bren 6164	. . 3 ⊢ (𝐶 ≈ 𝐷 ↔ ∃y y:𝐶–1-1-onto→𝐷)
3		eeanv 1804	. . . 4 ⊢ (∃x∃y(x:A–1-1-onto→B ∧ y:𝐶–1-1-onto→𝐷) ↔ (∃x x:A–1-1-onto→B ∧ ∃y y:𝐶–1-1-onto→𝐷))
4		vex 2554	. . . . . . . 8 ⊢ x ∈ V
5		vex 2554	. . . . . . . 8 ⊢ y ∈ V
6	4, 5	unex 4142	. . . . . . 7 ⊢ (x ∪ y) ∈ V
7		f1oun 5089	. . . . . . 7 ⊢ (((x:A–1-1-onto→B ∧ y:𝐶–1-1-onto→𝐷) ∧ ((A ∩ 𝐶) = ∅ ∧ (B ∩ 𝐷) = ∅)) → (x ∪ y):(A ∪ 𝐶)–1-1-onto→(B ∪ 𝐷))
8		f1oen3g 6170	. . . . . . 7 ⊢ (((x ∪ y) ∈ V ∧ (x ∪ y):(A ∪ 𝐶)–1-1-onto→(B ∪ 𝐷)) → (A ∪ 𝐶) ≈ (B ∪ 𝐷))
9	6, 7, 8	sylancr 393	. . . . . 6 ⊢ (((x:A–1-1-onto→B ∧ y:𝐶–1-1-onto→𝐷) ∧ ((A ∩ 𝐶) = ∅ ∧ (B ∩ 𝐷) = ∅)) → (A ∪ 𝐶) ≈ (B ∪ 𝐷))
10	9	ex 108	. . . . 5 ⊢ ((x:A–1-1-onto→B ∧ y:𝐶–1-1-onto→𝐷) → (((A ∩ 𝐶) = ∅ ∧ (B ∩ 𝐷) = ∅) → (A ∪ 𝐶) ≈ (B ∪ 𝐷)))
11	10	exlimivv 1773	. . . 4 ⊢ (∃x∃y(x:A–1-1-onto→B ∧ y:𝐶–1-1-onto→𝐷) → (((A ∩ 𝐶) = ∅ ∧ (B ∩ 𝐷) = ∅) → (A ∪ 𝐶) ≈ (B ∪ 𝐷)))
12	3, 11	sylbir 125	. . 3 ⊢ ((∃x x:A–1-1-onto→B ∧ ∃y y:𝐶–1-1-onto→𝐷) → (((A ∩ 𝐶) = ∅ ∧ (B ∩ 𝐷) = ∅) → (A ∪ 𝐶) ≈ (B ∪ 𝐷)))
13	1, 2, 12	syl2anb 275	. 2 ⊢ ((A ≈ B ∧ 𝐶 ≈ 𝐷) → (((A ∩ 𝐶) = ∅ ∧ (B ∩ 𝐷) = ∅) → (A ∪ 𝐶) ≈ (B ∪ 𝐷)))
14	13	imp 115	1 ⊢ (((A ≈ B ∧ 𝐶 ≈ 𝐷) ∧ ((A ∩ 𝐶) = ∅ ∧ (B ∩ 𝐷) = ∅)) → (A ∪ 𝐶) ≈ (B ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   ∪ cun 2909   ∩ cin 2910  ∅c0 3218   class class class wbr 3755  –1-1-onto→wf1o 4844   ≈ cen 6155

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-en 6158

This theorem is referenced by:  frecfzennn  8884