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Mirrors > Home > MPE Home > Th. List > ssequn2 | Structured version Visualization version GIF version |
Description: A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.) |
Ref | Expression |
---|---|
ssequn2 | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssequn1 4196 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵) | |
2 | uncom 4168 | . . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) | |
3 | 2 | eqeq1i 2740 | . 2 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) |
4 | 1, 3 | bitri 275 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∪ cun 3961 ⊆ wss 3963 |
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 df-ss 3980 |
This theorem is referenced by: unabs 4271 undifr 4489 tppreqb 4810 pwssun 5580 cnvimassrndm 6174 relresfld 6298 ordssun 6488 ordequn 6489 onunel 6491 onun2 6494 oneluni 6505 fsnunf 7205 sorpssun 7749 ordunpr 7846 omun 7910 fodomr 9167 unfi 9210 enp1ilem 9310 pwfilem 9354 fodomfir 9366 brwdom2 9611 sucprcreg 9639 dfacfin7 10437 hashbclem 14488 incexclem 15869 ramub1lem1 17060 ramub1lem2 17061 mreexmrid 17688 lspun0 21027 lbsextlem4 21181 cldlp 23174 ordtuni 23214 lfinun 23549 cldsubg 24135 trust 24254 nulmbl2 25585 limcmpt2 25934 cnplimc 25937 dvreslem 25959 dvaddbr 25989 dvmulbr 25990 dvmulbrOLD 25991 lhop 26070 plypf1 26266 coeeulem 26278 coeeu 26279 coef2 26285 rlimcnp 27023 noetalem1 27801 addsproplem2 28018 ex-un 30453 shs0i 31478 chj0i 31484 disjun0 32615 ffsrn 32747 difioo 32791 symgcom2 33087 eulerpartlemt 34353 fineqvac 35090 subfacp1lem1 35164 cvmscld 35258 mthmpps 35567 refssfne 36341 topjoin 36348 pibt2 37400 poimirlem3 37610 poimirlem28 37635 rntrclfvOAI 42679 istopclsd 42688 nacsfix 42700 diophrw 42747 tfsconcatb0 43334 onsucunipr 43362 oaun3 43372 clcnvlem 43613 cnvrcl0 43615 dmtrcl 43617 rntrcl 43618 iunrelexp0 43692 dmtrclfvRP 43720 rntrclfv 43722 cotrclrcl 43732 clsk3nimkb 44030 limciccioolb 45577 limcicciooub 45593 ioccncflimc 45841 icocncflimc 45845 stoweidlem44 46000 dirkercncflem3 46061 fourierdlem62 46124 ismeannd 46423 |
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