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Mirrors > Home > ILE Home > Th. List > eqsstri | GIF version |
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.) |
Ref | Expression |
---|---|
eqsstr.1 | ⊢ 𝐴 = 𝐵 |
eqsstr.2 | ⊢ 𝐵 ⊆ 𝐶 |
Ref | Expression |
---|---|
eqsstri | ⊢ 𝐴 ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsstr.2 | . 2 ⊢ 𝐵 ⊆ 𝐶 | |
2 | eqsstr.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 2 | sseq1i 3123 | . 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶) |
4 | 1, 3 | mpbir 145 | 1 ⊢ 𝐴 ⊆ 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ⊆ wss 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 |
This theorem is referenced by: eqsstrri 3130 ssrab2 3182 rabssab 3184 difdifdirss 3447 opabss 3992 brab2ga 4614 relopabi 4665 dmopabss 4751 resss 4843 relres 4847 exse2 4913 rnin 4948 rnxpss 4970 cnvcnvss 4993 dmmptss 5035 cocnvss 5064 fnres 5239 resasplitss 5302 fabexg 5310 f0 5313 ffvresb 5583 isoini2 5720 dmoprabss 5853 elmpocl 5968 tposssxp 6146 dftpos4 6160 smores 6189 smores2 6191 iordsmo 6194 swoer 6457 swoord1 6458 swoord2 6459 ecss 6470 ecopovsym 6525 ecopovtrn 6526 ecopover 6527 ecopovsymg 6528 ecopovtrng 6529 ecopoverg 6530 sbthlem7 6851 caserel 6972 ctssdccl 6996 pinn 7117 niex 7120 ltrelpi 7132 dmaddpi 7133 dmmulpi 7134 enqex 7168 ltrelnq 7173 enq0ex 7247 ltrelpr 7313 enrex 7545 ltrelsr 7546 ltrelre 7641 axaddf 7676 axmulf 7677 ltrelxr 7825 lerelxr 7827 nn0ssre 8981 nn0ssz 9072 rpre 9448 fz1ssfz0 9897 cau3 10887 fsum3cvg3 11165 isumshft 11259 explecnv 11274 clim2prod 11308 ntrivcvgap 11317 dvdszrcl 11498 dvdsflip 11549 infssuzcldc 11644 phimullem 11901 ctiunctlemuom 11949 structcnvcnv 11975 fvsetsid 11993 strleun 12048 dmtopon 12190 lmfval 12361 lmbrf 12384 cnconst2 12402 txuni2 12425 xmeter 12605 ivthinclemex 12789 dvrecap 12846 |
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