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Mirrors > Home > MPE Home > Th. List > eqimss2 | Structured version Visualization version GIF version |
Description: Equality implies inclusion. (Contributed by NM, 23-Nov-2003.) |
Ref | Expression |
---|---|
eqimss2 | ⊢ (𝐵 = 𝐴 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss 4053 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | 1 | eqcoms 2742 | 1 ⊢ (𝐵 = 𝐴 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ⊆ wss 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-cleq 2726 df-ss 3979 |
This theorem is referenced by: pweq 4618 ifpprsnss 4768 unieq 4922 disjeq2 5118 disjeq1 5121 poeq2 5600 freq2 5656 seeq1 5658 seeq2 5659 dmcoeq 5991 xp11 6196 suc11 6492 funeq 6587 fimadmfoALT 6831 foco 6834 fconst3 7232 sorpssuni 7750 sorpssint 7751 tposeq 8251 oaass 8597 odi 8615 oen0 8622 mapssfset 8889 inficl 9462 fodomfi2 10097 zorng 10541 rlimclim 15578 imasaddfnlem 17574 imasvscafn 17583 gasubg 19332 pgpssslw 19646 dprddisj2 20073 dprd2da 20076 imadrhmcl 20814 evlslem6 22122 topgele 22951 topontopn 22961 connima 23448 islocfin 23540 ptbasfi 23604 txdis 23655 neifil 23903 elfm3 23973 rnelfmlem 23975 alexsubALTlem3 24072 alexsubALTlem4 24073 utopsnneiplem 24271 lmclimf 25351 uniiccdif 25626 dv11cn 26054 plypf1 26265 2pthon3v 29972 hstoh 32260 dmdi2 32332 disjeq1f 32592 eulerpartlemd 34347 rrvdmss 34430 umgr2cycllem 35124 refssfne 36340 neibastop3 36344 topmeet 36346 topjoin 36347 fnemeet2 36349 fnejoin1 36350 bj-restuni 37079 bj-inexeqex 37136 bj-idreseq 37144 heiborlem3 37799 funALTVeq 38681 disjeq 38715 lsatelbN 38987 lkrscss 39079 lshpset2N 39100 mapdrvallem2 41627 hdmaprnlem3eN 41840 hdmaplkr 41895 uneqsn 44014 ssrecnpr 44303 founiiun 45121 founiiun0 45132 caragendifcl 46469 fnfocofob 47028 imasetpreimafvbijlemfo 47329 unilbeu 48773 |
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