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Mirrors > Home > NFE Home > Th. List > snelpw1 | GIF version |
Description: Membership of a singleton in a unit power class. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
snelpw1 | ⊢ ({A} ∈ ℘1B ↔ A ∈ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2355 | . . . 4 ⊢ ({A} = {x} ↔ {x} = {A}) | |
2 | vex 2862 | . . . . 5 ⊢ x ∈ V | |
3 | 2 | sneqb 3876 | . . . 4 ⊢ ({x} = {A} ↔ x = A) |
4 | 1, 3 | bitri 240 | . . 3 ⊢ ({A} = {x} ↔ x = A) |
5 | 4 | rexbii 2639 | . 2 ⊢ (∃x ∈ B {A} = {x} ↔ ∃x ∈ B x = A) |
6 | elpw1 4144 | . 2 ⊢ ({A} ∈ ℘1B ↔ ∃x ∈ B {A} = {x}) | |
7 | risset 2661 | . 2 ⊢ (A ∈ B ↔ ∃x ∈ B x = A) | |
8 | 5, 6, 7 | 3bitr4i 268 | 1 ⊢ ({A} ∈ ℘1B ↔ A ∈ B) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 {csn 3737 ℘1cpw1 4135 |
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 4078 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 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 2478 df-ne 2518 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 df-1c 4136 df-pw1 4137 |
This theorem is referenced by: eqpw1 4162 pw1in 4164 pw10b 4166 pw1disj 4167 pw111 4170 ins2kss 4279 ins3kss 4280 ins2kexg 4305 ins3kexg 4306 dfpw2 4327 eqpw1uni 4330 ssfin 4470 ncfinraiselem2 4480 ncfinraise 4481 ncfinlower 4483 tfinsuc 4498 nnadjoinlem1 4519 sfindbl 4530 tfinnnlem1 4533 enpw1 6062 enprmaplem4 6079 nenpw1pwlem2 6085 nchoicelem11 6299 scancan 6331 |
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