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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 2828 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | orbi1d 916 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) |
3 | elun 4163 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐶) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶)) | |
4 | elun 4163 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) | |
5 | 2, 3, 4 | 3bitr4g 314 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ (𝐴 ∪ 𝐶) ↔ 𝑥 ∈ (𝐵 ∪ 𝐶))) |
6 | 5 | eqrdv 2733 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 |
This theorem is referenced by: uneq2 4172 uneq12 4173 uneq1i 4174 uneq1d 4177 unineq 4294 prprc1 4770 relresfld 6298 unexbOLD 7767 oarec 8599 xpider 8827 ralxpmap 8935 undifixp 8973 findcard2 9203 unxpdom 9287 enp1ilem 9310 pwfilem 9354 domunfican 9359 fin1a2lem10 10447 incexclem 15869 lcmfunsnlem 16675 ramub1lem1 17060 ramub1 17062 mreexexlem3d 17691 mreexexlem4d 17692 ipodrsima 18599 mplsubglem 22037 mretopd 23116 iscldtop 23119 nconnsubb 23447 plyval 26247 spanun 31574 difeq 32546 unelldsys 34139 isros 34149 unelros 34152 difelros 34153 rossros 34161 measun 34192 inelcarsg 34293 actfunsnf1o 34598 actfunsnrndisj 34599 mrsubvrs 35507 altopthsn 35943 rankung 36148 bj-adjg1 37026 poimirlem28 37635 islshp 38961 lshpset2N 39101 paddval 39781 nacsfix 42700 eldioph4b 42799 eldioph4i 42800 diophren 42801 clsk3nimkb 44030 isotone1 44038 fiiuncl 45005 founiiun0 45133 infxrpnf 45396 meadjun 46418 hoidmvle 46556 |
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