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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 |
---|---|
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 3940 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
4 | nfra1 3270 | . 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 | |
5 | 3, 4 | nfxfr 1856 | 1 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2888 ∀wral 3065 ⊆ wss 3915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-v 3450 df-in 3922 df-ss 3932 |
This theorem is referenced by: ssrexf 4013 nfpw 4584 ssiun2s 5013 triun 5242 iunopeqop 5483 ssopab2bw 5509 ssopab2b 5511 nffr 5612 nfrel 5740 nffun 6529 nff 6669 fvmptss 6965 ssoprab2b 7431 eqoprab2bw 7432 tfis 7796 ovmptss 8030 nffrecs 8219 nfwrecsOLD 8253 oawordeulem 8506 nnawordex 8589 r1val1 9729 cardaleph 10032 nfsum1 15581 nfsum 15582 nfsumOLD 15583 nfcprod1 15800 nfcprod 15801 iunconn 22795 ovolfiniun 24881 ovoliunlem3 24884 ovoliun 24885 ovoliun2 24886 ovoliunnul 24887 limciun 25274 ssiun2sf 31520 ssrelf 31576 funimass4f 31593 fsumiunle 31767 prodindf 32662 esumiun 32733 bnj1408 33688 totbndbnd 36277 naddwordnexlem4 41747 ss2iundf 42005 iunconnlem2 43291 iinssdf 43423 rnmptssbi 43563 stoweidlem53 44368 stoweidlem57 44372 meaiunincf 44798 meaiuninc3 44800 opnvonmbllem2 44948 smflim 45092 nfsetrecs 47205 setrec2fun 47211 |
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