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Mirrors > Home > ILE Home > Th. List > nfss | GIF version |
Description: If 𝑥 is not free in 𝐴 and 𝐵, it is not free in 𝐴 ⊆ 𝐵. (Contributed by NM, 27-Dec-1996.) |
Ref | Expression |
---|---|
dfss2f.1 | ⊢ Ⅎ𝑥𝐴 |
dfss2f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfss | ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | dfss2f.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | dfss3f 3089 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
4 | nfra1 2466 | . 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 | |
5 | 3, 4 | nfxfr 1450 | 1 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnf 1436 ∈ wcel 1480 Ⅎwnfc 2268 ∀wral 2416 ⊆ 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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 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-nfc 2270 df-ral 2421 df-in 3077 df-ss 3084 |
This theorem is referenced by: ssrexf 3159 nfpw 3523 ssiun2s 3857 triun 4039 ssopab2b 4198 nffrfor 4270 tfis 4497 nfrel 4624 nffun 5146 nff 5269 fvmptssdm 5505 ssoprab2b 5828 nfsum1 11125 nfsum 11126 nfcprod1 11323 nfcprod 11324 |
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