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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 2898 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | orbi1d 910 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) |
3 | elun 4122 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐶) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶)) | |
4 | elun 4122 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) | |
5 | 2, 3, 4 | 3bitr4g 315 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ (𝐴 ∪ 𝐶) ↔ 𝑥 ∈ (𝐵 ∪ 𝐶))) |
6 | 5 | eqrdv 2816 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-un 3938 |
This theorem is referenced by: uneq2 4130 uneq12 4131 uneq1i 4132 uneq1d 4135 unineq 4251 prprc1 4693 uniprg 4844 relresfld 6120 unexb 7460 oarec 8177 xpider 8357 ralxpmap 8448 undifixp 8486 unxpdom 8713 enp1ilem 8740 findcard2 8746 domunfican 8779 pwfilem 8806 fin1a2lem10 9819 incexclem 15179 lcmfunsnlem 15973 ramub1lem1 16350 ramub1 16352 mreexexlem3d 16905 mreexexlem4d 16906 ipodrsima 17763 mplsubglem 20142 mretopd 21628 iscldtop 21631 nconnsubb 21959 plyval 24710 spanun 29249 difeq 30207 unelldsys 31316 isros 31326 unelros 31329 difelros 31330 rossros 31338 measun 31369 inelcarsg 31468 actfunsnf1o 31774 actfunsnrndisj 31775 mrsubvrs 32666 altopthsn 33319 rankung 33524 poimirlem28 34801 islshp 35995 lshpset2N 36135 paddval 36814 nacsfix 39187 eldioph4b 39286 eldioph4i 39287 diophren 39288 clsk3nimkb 40268 isotone1 40276 fiiuncl 41204 founiiun0 41327 infxrpnf 41597 meadjun 42621 hoidmvle 42759 |
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