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Mirrors > Home > ILE Home > Th. List > unen | GIF version |
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.) |
Ref | Expression |
---|---|
unen | ⊢ (((A ≈ B ∧ 𝐶 ≈ 𝐷) ∧ ((A ∩ 𝐶) = ∅ ∧ (B ∩ 𝐷) = ∅)) → (A ∪ 𝐶) ≈ (B ∪ 𝐷)) |
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)) |
Colors of variables: wff set class |
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 |
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