Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2604 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)) | |
2 | 1 | abbidv 2235 | . 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}) |
3 | dfuni2 3708 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
4 | dfuni2 3708 | . 2 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} | |
5 | 2, 3, 4 | 3eqtr4g 2175 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 {cab 2103 ∃wrex 2394 ∪ cuni 3706 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-uni 3707 |
This theorem is referenced by: unieqi 3716 unieqd 3717 uniintsnr 3777 iununir 3866 treq 4002 limeq 4269 uniex 4329 uniexg 4331 ordsucunielexmid 4416 onsucuni2 4449 nnpredcl 4506 elvvuni 4573 unielrel 5036 unixp0im 5045 iotass 5075 nnsucuniel 6359 en1bg 6662 omp1eom 6948 ctmlemr 6961 uniopn 12095 istopon 12107 eltg3 12153 tgdom 12168 cldval 12195 ntrfval 12196 clsfval 12197 neifval 12236 tgrest 12265 cnprcl2k 12302 bj-uniex 13042 bj-uniexg 13043 nnsf 13126 peano3nninf 13128 exmidsbthr 13145 |
Copyright terms: Public domain | W3C validator |