Theorem unen 6818

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 6749	. . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑥 𝑥:𝐴–1-1-onto→𝐵)
2		bren 6749	. . 3 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑦 𝑦:𝐶–1-1-onto→𝐷)
3		eeanv 1932	. . . 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 2742	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 2742	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	4, 5	unex 4443	. . . . . . 7 ⊢ (𝑥 ∪ 𝑦) ∈ V
7		f1oun 5483	. . . . . . 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 6756	. . . . . . 7 ⊢ (((𝑥 ∪ 𝑦) ∈ V ∧ (𝑥 ∪ 𝑦):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))
9	6, 7, 8	sylancr 414	. . . . . 6 ⊢ (((𝑥:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))
10	9	ex 115	. . . . 5 ⊢ ((𝑥:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐶–1-1-onto→𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
11	10	exlimivv 1896	. . . 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 135	. . 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 291	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
14	13	imp 124	1 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353  ∃wex 1492   ∈ wcel 2148  Vcvv 2739   ∪ cun 3129   ∩ cin 3130  ∅c0 3424   class class class wbr 4005  –1-1-onto→wf1o 5217   ≈ cen 6740

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-en 6743

This theorem is referenced by:  enpr2d  6819  phplem2  6855  fiunsnnn  6883  unsnfi  6920  endjusym  7097  pm54.43  7191  endjudisj  7211  djuen  7212  frecfzennn  10428  unennn  12400