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Mirrors > Home > MPE Home > Th. List > syl5eqssr | Structured version Visualization version GIF version |
Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
syl5eqssr.1 | ⊢ 𝐵 = 𝐴 |
syl5eqssr.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
syl5eqssr | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl5eqssr.1 | . . 3 ⊢ 𝐵 = 𝐴 | |
2 | 1 | eqcomi 2619 | . 2 ⊢ 𝐴 = 𝐵 |
3 | syl5eqssr.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
4 | 2, 3 | syl5eqss 3612 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ⊆ wss 3540 |
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-in 3547 df-ss 3554 |
This theorem is referenced by: relcnvtr 5572 fimacnvdisj 5996 dffv2 6181 fimacnv 6255 f1ompt 6290 fnwelem 7179 tfrlem15 7375 omxpenlem 7946 hartogslem1 8330 infxpidm2 8723 alephgeom 8788 infenaleph 8797 cfflb 8964 pwfseqlem5 9364 imasvscafn 16020 mrieqvlemd 16112 cnvps 17035 dirdm 17057 tsrdir 17061 frmdss2 17223 iinopn 20532 neitr 20794 xkococnlem 21272 tgpconcomp 21726 trcfilu 21908 mbfconstlem 23202 itg2seq 23315 limcdif 23446 dvres2lem 23480 c1lip3 23566 lhop 23583 plyeq0 23771 dchrghm 24781 chssoc 27739 hauseqcn 29269 carsgclctunlem3 29709 cvmliftmolem1 30517 cvmlift2lem9a 30539 cvmlift2lem9 30547 cnres2 32732 rngunsnply 36762 proot1hash 36797 clcnvlem 36949 cnvtrcl0 36952 trrelsuperrel2dg 36982 brtrclfv2 37038 fourierdlem92 39091 umgrres1 40533 umgr2v2e 40741 vsetrec 42245 |
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