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Mirrors > Home > NFE Home > Th. List > siex | GIF version |
Description: The singleton image of a set is a set. (Contributed by SF, 7-Jan-2015.) |
Ref | Expression |
---|---|
siex.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
siex | ⊢ SI A ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | siex.1 | . 2 ⊢ A ∈ V | |
2 | siexg 4753 | . 2 ⊢ (A ∈ V → SI A ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ SI A ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1710 Vcvv 2860 SI csi 4721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-addc 4379 df-nnc 4380 df-phi 4566 df-op 4567 df-opab 4624 df-br 4641 df-si 4729 |
This theorem is referenced by: si3ex 5807 pw1fnex 5853 refex 5912 enpw1lem1 6062 enpw1 6063 enprmaplem1 6077 lecex 6116 ceex 6175 tcfnex 6245 nmembers1lem1 6269 nchoicelem11 6300 nchoicelem16 6305 |
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