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Mirrors > Home > ILE Home > Th. List > unieq | GIF version |
Description: Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
unieq | ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexeq 2550 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)) | |
2 | 1 | abbidv 2196 | . 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}) |
3 | dfuni2 3603 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
4 | dfuni2 3603 | . 2 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} | |
5 | 2, 3, 4 | 3eqtr4g 2138 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 {cab 2067 ∃wrex 2349 ∪ cuni 3601 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-uni 3602 |
This theorem is referenced by: unieqi 3611 unieqd 3612 uniintsnr 3672 iununir 3759 treq 3881 limeq 4132 uniex 4192 uniexg 4193 ordsucunielexmid 4274 onsucuni2 4307 elvvuni 4422 unielrel 4865 unixp0im 4874 iotass 4904 en1bg 6303 bj-uniex 10708 bj-uniexg 10709 |
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