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Mirrors > Home > NFE Home > Th. List > elpw121c | GIF version |
Description: Membership in ℘1℘11c (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
elpw121c | ⊢ (A ∈ ℘1℘11c ↔ ∃x A = {{{x}}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpw1 4144 | . 2 ⊢ (A ∈ ℘1℘11c ↔ ∃y ∈ ℘1 1cA = {y}) | |
2 | df-rex 2620 | . . . 4 ⊢ (∃y ∈ ℘1 1cA = {y} ↔ ∃y(y ∈ ℘11c ∧ A = {y})) | |
3 | elpw11c 4147 | . . . . . . 7 ⊢ (y ∈ ℘11c ↔ ∃x y = {{x}}) | |
4 | 3 | anbi1i 676 | . . . . . 6 ⊢ ((y ∈ ℘11c ∧ A = {y}) ↔ (∃x y = {{x}} ∧ A = {y})) |
5 | 19.41v 1901 | . . . . . 6 ⊢ (∃x(y = {{x}} ∧ A = {y}) ↔ (∃x y = {{x}} ∧ A = {y})) | |
6 | 4, 5 | bitr4i 243 | . . . . 5 ⊢ ((y ∈ ℘11c ∧ A = {y}) ↔ ∃x(y = {{x}} ∧ A = {y})) |
7 | 6 | exbii 1582 | . . . 4 ⊢ (∃y(y ∈ ℘11c ∧ A = {y}) ↔ ∃y∃x(y = {{x}} ∧ A = {y})) |
8 | 2, 7 | bitri 240 | . . 3 ⊢ (∃y ∈ ℘1 1cA = {y} ↔ ∃y∃x(y = {{x}} ∧ A = {y})) |
9 | excom 1741 | . . . 4 ⊢ (∃y∃x(y = {{x}} ∧ A = {y}) ↔ ∃x∃y(y = {{x}} ∧ A = {y})) | |
10 | snex 4111 | . . . . . 6 ⊢ {{x}} ∈ V | |
11 | sneq 3744 | . . . . . . 7 ⊢ (y = {{x}} → {y} = {{{x}}}) | |
12 | 11 | eqeq2d 2364 | . . . . . 6 ⊢ (y = {{x}} → (A = {y} ↔ A = {{{x}}})) |
13 | 10, 12 | ceqsexv 2894 | . . . . 5 ⊢ (∃y(y = {{x}} ∧ A = {y}) ↔ A = {{{x}}}) |
14 | 13 | exbii 1582 | . . . 4 ⊢ (∃x∃y(y = {{x}} ∧ A = {y}) ↔ ∃x A = {{{x}}}) |
15 | 9, 14 | bitri 240 | . . 3 ⊢ (∃y∃x(y = {{x}} ∧ A = {y}) ↔ ∃x A = {{{x}}}) |
16 | 8, 15 | bitri 240 | . 2 ⊢ (∃y ∈ ℘1 1cA = {y} ↔ ∃x A = {{{x}}}) |
17 | 1, 16 | bitri 240 | 1 ⊢ (A ∈ ℘1℘11c ↔ ∃x A = {{{x}}}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 {csn 3737 1cc1c 4134 ℘1cpw1 4135 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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: elpw131c 4149 opkelimagekg 4271 ndisjrelk 4323 eqpwrelk 4478 ncfinraiselem2 4480 ncfinlowerlem1 4482 eqtfinrelk 4486 evenfinex 4503 oddfinex 4504 evenodddisjlem1 4515 nnadjoinlem1 4519 nnpweqlem1 4522 srelk 4524 tfinnnlem1 4533 spfinex 4537 dfop2lem1 4573 setconslem2 4732 |
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