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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 2677 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | orbi1d 735 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) |
3 | elun 3715 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐶) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶)) | |
4 | elun 3715 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) | |
5 | 2, 3, 4 | 3bitr4g 302 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ (𝐴 ∪ 𝐶) ↔ 𝑥 ∈ (𝐵 ∪ 𝐶))) |
6 | 5 | eqrdv 2608 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 |
This theorem is referenced by: uneq2 3723 uneq12 3724 uneq1i 3725 uneq1d 3728 unineq 3836 prprc1 4243 uniprg 4386 relresfld 5579 unexb 6856 oarec 7529 xpider 7705 ralxpmap 7793 undifixp 7830 unxpdom 8052 enp1ilem 8079 findcard2 8085 domunfican 8118 pwfilem 8143 fin1a2lem10 9114 incexclem 14407 lcmfunsnlem 15192 ramub1lem1 15568 ramub1 15570 mreexexlem3d 16129 mreexexlem4d 16130 ipodrsima 16988 mplsubglem 19255 mretopd 20706 iscldtop 20709 nconsubb 21036 plyval 23753 spanun 27788 difeq 28739 unelldsys 29548 isros 29558 unelros 29561 difelros 29562 rossros 29570 measun 29601 inelcarsg 29700 mrsubvrs 30673 nofulllem2 31102 altopthsn 31238 rankung 31443 poimirlem28 32607 islshp 33284 lshpset2N 33424 paddval 34102 nacsfix 36293 eldioph4b 36393 eldioph4i 36394 diophren 36395 clsk3nimkb 37358 isotone1 37366 compne 37665 fiiuncl 38259 founiiun0 38372 meadjun 39355 hoidmvle 39490 |
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