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Theorem pwsnALT 4402
Description: Alternate proof of pwsn 4401, more direct. (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
pwsnALT 𝒫 {𝐴} = {∅, {𝐴}}

Proof of Theorem pwsnALT
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfss2 3577 . . . . . . . . 9 (𝑥 ⊆ {𝐴} ↔ ∀𝑦(𝑦𝑥𝑦 ∈ {𝐴}))
2 velsn 4169 . . . . . . . . . . 11 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
32imbi2i 326 . . . . . . . . . 10 ((𝑦𝑥𝑦 ∈ {𝐴}) ↔ (𝑦𝑥𝑦 = 𝐴))
43albii 1744 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 ∈ {𝐴}) ↔ ∀𝑦(𝑦𝑥𝑦 = 𝐴))
51, 4bitri 264 . . . . . . . 8 (𝑥 ⊆ {𝐴} ↔ ∀𝑦(𝑦𝑥𝑦 = 𝐴))
6 neq0 3911 . . . . . . . . . 10 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
7 exintr 1821 . . . . . . . . . 10 (∀𝑦(𝑦𝑥𝑦 = 𝐴) → (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥𝑦 = 𝐴)))
86, 7syl5bi 232 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 = 𝐴) → (¬ 𝑥 = ∅ → ∃𝑦(𝑦𝑥𝑦 = 𝐴)))
9 df-clel 2622 . . . . . . . . . . 11 (𝐴𝑥 ↔ ∃𝑦(𝑦 = 𝐴𝑦𝑥))
10 exancom 1785 . . . . . . . . . . 11 (∃𝑦(𝑦 = 𝐴𝑦𝑥) ↔ ∃𝑦(𝑦𝑥𝑦 = 𝐴))
119, 10bitr2i 265 . . . . . . . . . 10 (∃𝑦(𝑦𝑥𝑦 = 𝐴) ↔ 𝐴𝑥)
12 snssi 4313 . . . . . . . . . 10 (𝐴𝑥 → {𝐴} ⊆ 𝑥)
1311, 12sylbi 207 . . . . . . . . 9 (∃𝑦(𝑦𝑥𝑦 = 𝐴) → {𝐴} ⊆ 𝑥)
148, 13syl6 35 . . . . . . . 8 (∀𝑦(𝑦𝑥𝑦 = 𝐴) → (¬ 𝑥 = ∅ → {𝐴} ⊆ 𝑥))
155, 14sylbi 207 . . . . . . 7 (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → {𝐴} ⊆ 𝑥))
1615anc2li 579 . . . . . 6 (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → (𝑥 ⊆ {𝐴} ∧ {𝐴} ⊆ 𝑥)))
17 eqss 3603 . . . . . 6 (𝑥 = {𝐴} ↔ (𝑥 ⊆ {𝐴} ∧ {𝐴} ⊆ 𝑥))
1816, 17syl6ibr 242 . . . . 5 (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → 𝑥 = {𝐴}))
1918orrd 393 . . . 4 (𝑥 ⊆ {𝐴} → (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
20 0ss 3949 . . . . . 6 ∅ ⊆ {𝐴}
21 sseq1 3610 . . . . . 6 (𝑥 = ∅ → (𝑥 ⊆ {𝐴} ↔ ∅ ⊆ {𝐴}))
2220, 21mpbiri 248 . . . . 5 (𝑥 = ∅ → 𝑥 ⊆ {𝐴})
23 eqimss 3641 . . . . 5 (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
2422, 23jaoi 394 . . . 4 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
2519, 24impbii 199 . . 3 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
2625abbii 2742 . 2 {𝑥𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
27 df-pw 4137 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
28 dfpr2 4171 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
2926, 27, 283eqtr4i 2658 1 𝒫 {𝐴} = {∅, {𝐴}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1992  {cab 2612  wss 3560  c0 3896  𝒫 cpw 4135  {csn 4153  {cpr 4155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-pw 4137  df-sn 4154  df-pr 4156
This theorem is referenced by: (None)
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