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Mirrors > Home > NFE Home > Th. List > elpw131c | GIF version |
Description: Membership in ℘1℘1℘11c. (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
elpw131c | ⊢ (A ∈ ℘1℘1℘11c ↔ ∃x A = {{{{x}}}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpw1 4144 | . 2 ⊢ (A ∈ ℘1℘1℘11c ↔ ∃y ∈ ℘1 ℘11cA = {y}) | |
2 | df-rex 2620 | . . . 4 ⊢ (∃y ∈ ℘1 ℘11cA = {y} ↔ ∃y(y ∈ ℘1℘11c ∧ A = {y})) | |
3 | elpw121c 4148 | . . . . . . 7 ⊢ (y ∈ ℘1℘11c ↔ ∃x y = {{{x}}}) | |
4 | 3 | anbi1i 676 | . . . . . 6 ⊢ ((y ∈ ℘1℘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 ∈ ℘1℘11c ∧ A = {y}) ↔ ∃x(y = {{{x}}} ∧ A = {y})) |
7 | 6 | exbii 1582 | . . . 4 ⊢ (∃y(y ∈ ℘1℘11c ∧ A = {y}) ↔ ∃y∃x(y = {{{x}}} ∧ A = {y})) |
8 | 2, 7 | bitri 240 | . . 3 ⊢ (∃y ∈ ℘1 ℘11cA = {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 ℘11cA = {y} ↔ ∃x A = {{{{x}}}}) |
17 | 1, 16 | bitri 240 | 1 ⊢ (A ∈ ℘1℘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-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: elpw141c 4150 nnsucelrlem1 4424 ltfinex 4464 eqpw1relk 4479 ncfinraiselem2 4480 ncfinlowerlem1 4482 eqtfinrelk 4486 oddfinex 4504 nnpweqlem1 4522 srelk 4524 |
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