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Mirrors > Home > MPE Home > Th. List > uneq1 | Structured version Visualization version GIF version |
Description: Equality theorem for the union of two classes. (Contributed by NM, 15-Jul-1993.) |
Ref | Expression |
---|---|
uneq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2820 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | orbi1d 913 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) |
3 | elun 4149 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐶) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶)) | |
4 | elun 4149 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) | |
5 | 2, 3, 4 | 3bitr4g 313 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ (𝐴 ∪ 𝐶) ↔ 𝑥 ∈ (𝐵 ∪ 𝐶))) |
6 | 5 | eqrdv 2728 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1539 ∈ wcel 2104 ∪ cun 3947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-un 3954 |
This theorem is referenced by: uneq2 4158 uneq12 4159 uneq1i 4160 uneq1d 4163 unineq 4278 prprc1 4770 uniprgOLD 4929 relresfld 6276 unexb 7739 oarec 8566 xpider 8786 ralxpmap 8894 undifixp 8932 findcard2 9168 pwfilem 9181 unxpdom 9257 enp1ilem 9282 findcard2OLD 9288 domunfican 9324 pwfilemOLD 9350 fin1a2lem10 10408 incexclem 15788 lcmfunsnlem 16584 ramub1lem1 16965 ramub1 16967 mreexexlem3d 17596 mreexexlem4d 17597 ipodrsima 18500 mplsubglem 21779 mretopd 22818 iscldtop 22821 nconnsubb 23149 plyval 25941 spanun 31063 difeq 32021 unelldsys 33452 isros 33462 unelros 33465 difelros 33466 rossros 33474 measun 33505 inelcarsg 33606 actfunsnf1o 33912 actfunsnrndisj 33913 mrsubvrs 34809 altopthsn 35235 rankung 35440 bj-adjg1 36229 poimirlem28 36821 islshp 38154 lshpset2N 38294 paddval 38974 nacsfix 41754 eldioph4b 41853 eldioph4i 41854 diophren 41855 clsk3nimkb 43095 isotone1 43103 fiiuncl 44055 founiiun0 44189 infxrpnf 44456 meadjun 45478 hoidmvle 45616 |
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