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Mirrors > Home > ILE Home > Th. List > uneq1 | GIF version |
Description: Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
uneq1 | ⊢ (A = B → (A ∪ 𝐶) = (B ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2098 | . . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B)) | |
2 | 1 | orbi1d 704 | . . 3 ⊢ (A = B → ((x ∈ A ∨ x ∈ 𝐶) ↔ (x ∈ B ∨ x ∈ 𝐶))) |
3 | elun 3078 | . . 3 ⊢ (x ∈ (A ∪ 𝐶) ↔ (x ∈ A ∨ x ∈ 𝐶)) | |
4 | elun 3078 | . . 3 ⊢ (x ∈ (B ∪ 𝐶) ↔ (x ∈ B ∨ x ∈ 𝐶)) | |
5 | 2, 3, 4 | 3bitr4g 212 | . 2 ⊢ (A = B → (x ∈ (A ∪ 𝐶) ↔ x ∈ (B ∪ 𝐶))) |
6 | 5 | eqrdv 2035 | 1 ⊢ (A = B → (A ∪ 𝐶) = (B ∪ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 628 = wceq 1242 ∈ wcel 1390 ∪ cun 2909 |
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-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-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 |
This theorem is referenced by: uneq2 3085 uneq12 3086 uneq1i 3087 uneq1d 3090 prprc1 3469 uniprg 3586 unexb 4143 relresfld 4790 relcoi1 4792 rdgeq2 5899 xpiderm 6113 bdunexb 9375 bj-unexg 9376 |
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