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Mirrors > Home > ILE Home > Th. List > sseq1 | GIF version |
Description: Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Ref | Expression |
---|---|
sseq1 | ⊢ (A = B → (A ⊆ 𝐶 ↔ B ⊆ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqss 2954 | . 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A)) | |
2 | sstr2 2946 | . . . 4 ⊢ (B ⊆ A → (A ⊆ 𝐶 → B ⊆ 𝐶)) | |
3 | 2 | adantl 262 | . . 3 ⊢ ((A ⊆ B ∧ B ⊆ A) → (A ⊆ 𝐶 → B ⊆ 𝐶)) |
4 | sstr2 2946 | . . . 4 ⊢ (A ⊆ B → (B ⊆ 𝐶 → A ⊆ 𝐶)) | |
5 | 4 | adantr 261 | . . 3 ⊢ ((A ⊆ B ∧ B ⊆ A) → (B ⊆ 𝐶 → A ⊆ 𝐶)) |
6 | 3, 5 | impbid 120 | . 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → (A ⊆ 𝐶 ↔ B ⊆ 𝐶)) |
7 | 1, 6 | sylbi 114 | 1 ⊢ (A = B → (A ⊆ 𝐶 ↔ B ⊆ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ⊆ wss 2911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-in 2918 df-ss 2925 |
This theorem is referenced by: sseq12 2962 sseq1i 2963 sseq1d 2966 nssne2 2996 psseq1 3025 sspsstr 3044 sbss 3323 pwjust 3352 elpw 3357 elpwg 3359 sssnr 3515 ssprr 3518 sstpr 3519 unimax 3605 trss 3854 elssabg 3893 bnd2 3917 mss 3953 exss 3954 ordtri2orexmid 4211 onsucsssucexmid 4212 sucprcreg 4227 tfis 4249 tfisi 4253 elnn 4271 nnregexmid 4285 releq 4365 xpsspw 4393 iss 4597 relcnvtr 4783 iotass 4827 fununi 4910 funcnvuni 4911 funimaexglem 4925 ffoss 5101 ssimaex 5177 tfrlem1 5864 nnsucsssuc 6010 qsss 6101 ssfiexmid 6254 elinp 6457 bj-om 9396 bj-2inf 9397 bj-nntrans 9411 bj-omtrans 9416 |
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