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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 2673 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)) | |
2 | 1 | abbidv 2295 | . 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}) |
3 | dfuni2 3811 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
4 | dfuni2 3811 | . 2 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} | |
5 | 2, 3, 4 | 3eqtr4g 2235 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 {cab 2163 ∃wrex 2456 ∪ cuni 3809 |
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-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-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-uni 3810 |
This theorem is referenced by: unieqi 3819 unieqd 3820 uniintsnr 3880 iununir 3970 treq 4107 limeq 4377 uniex 4437 uniexg 4439 ordsucunielexmid 4530 onsucuni2 4563 nnpredcl 4622 elvvuni 4690 unielrel 5156 unixp0im 5165 iotass 5195 nnsucuniel 6495 en1bg 6799 omp1eom 7093 ctmlemr 7106 nnnninfeq2 7126 uniopn 13471 istopon 13483 eltg3 13527 tgdom 13542 cldval 13569 ntrfval 13570 clsfval 13571 neifval 13610 tgrest 13639 cnprcl2k 13676 bj-uniex 14639 bj-uniexg 14640 nnsf 14724 peano3nninf 14726 exmidsbthr 14741 |
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