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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 2660 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)) | |
2 | 1 | abbidv 2282 | . 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}) |
3 | dfuni2 3785 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
4 | dfuni2 3785 | . 2 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} | |
5 | 2, 3, 4 | 3eqtr4g 2222 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 {cab 2150 ∃wrex 2443 ∪ cuni 3783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-uni 3784 |
This theorem is referenced by: unieqi 3793 unieqd 3794 uniintsnr 3854 iununir 3943 treq 4080 limeq 4349 uniex 4409 uniexg 4411 ordsucunielexmid 4502 onsucuni2 4535 nnpredcl 4594 elvvuni 4662 unielrel 5125 unixp0im 5134 iotass 5164 nnsucuniel 6454 en1bg 6757 omp1eom 7051 ctmlemr 7064 nnnninfeq2 7084 uniopn 12546 istopon 12558 eltg3 12604 tgdom 12619 cldval 12646 ntrfval 12647 clsfval 12648 neifval 12687 tgrest 12716 cnprcl2k 12753 bj-uniex 13640 bj-uniexg 13641 nnsf 13726 peano3nninf 13728 exmidsbthr 13743 |
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