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Mirrors > Home > MPE Home > Th. List > snssg | Structured version Visualization version GIF version |
Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) |
Ref | Expression |
---|---|
snssg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2676 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
2 | sneq 4135 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
3 | 2 | sseq1d 3595 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ⊆ 𝐵 ↔ {𝐴} ⊆ 𝐵)) |
4 | vex 3176 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 4 | snss 4259 | . 2 ⊢ (𝑥 ∈ 𝐵 ↔ {𝑥} ⊆ 𝐵) |
6 | 1, 3, 5 | vtoclbg 3240 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-sn 4126 |
This theorem is referenced by: tppreqb 4277 snssi 4280 prssg 4290 fvimacnvALT 6244 fr3nr 6871 vdwapid1 15517 acsfn 16143 cycsubg2 17454 cycsubg2cl 17455 pgpfac1lem1 18296 pgpfac1lem3a 18298 pgpfac1lem3 18299 pgpfac1lem5 18301 pgpfaclem2 18304 lspsnid 18814 lidldvgen 19076 isneip 20719 elnei 20725 iscnp4 20877 cnpnei 20878 nlly2i 21089 1stckgenlem 21166 flimopn 21589 flimclslem 21598 fclsneii 21631 fcfnei 21649 limcvallem 23441 ellimc2 23447 limcflf 23451 limccnp 23461 limccnp2 23462 limcco 23463 lhop2 23582 plyrem 23864 isppw 24640 lpvtx 25734 h1did 27794 erdszelem8 30434 neibastop2 31526 prnc 33036 proot1mul 36796 uneqsn 37341 islptre 38686 rrxsnicc 39196 |
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