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Mirrors > Home > MPE Home > Th. List > nfss | Structured version Visualization version 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 |
---|---|
dfssf.1 | ⊢ Ⅎ𝑥𝐴 |
dfssf.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfss | ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfssf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | dfssf.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | dfss3f 3987 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
4 | nfra1 3282 | . 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 | |
5 | 3, 4 | nfxfr 1850 | 1 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ∀wral 3059 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-10 2139 ax-11 2155 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-clel 2814 df-nfc 2890 df-ral 3060 df-ss 3980 |
This theorem is referenced by: ssrexf 4062 nfpw 4624 ssiun2s 5053 triun 5280 iunopeqop 5531 ssopab2bw 5557 ssopab2b 5559 nffr 5662 nfrel 5792 nffun 6591 nff 6733 fvmptss 7028 ssoprab2b 7502 eqoprab2bw 7503 tfis 7876 ovmptss 8117 nffrecs 8307 nfwrecsOLD 8341 oawordeulem 8591 nnawordex 8674 r1val1 9824 cardaleph 10127 nfsum1 15723 nfsum 15724 nfcprod1 15941 nfcprod 15942 iunconn 23452 ovolfiniun 25550 ovoliunlem3 25553 ovoliun 25554 ovoliun2 25555 ovoliunnul 25556 limciun 25944 ssiun2sf 32580 ssrelf 32635 funimass4f 32654 fsumiunle 32836 prodindf 34004 esumiun 34075 bnj1408 35029 totbndbnd 37776 naddwordnexlem4 43391 ss2iundf 43649 iunconnlem2 44933 iinssdf 45079 rnmptssbi 45206 stoweidlem53 46009 stoweidlem57 46013 meaiunincf 46439 meaiuninc3 46441 opnvonmbllem2 46589 smflim 46733 nfsetrecs 48917 setrec2fun 48923 |
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