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Theorem eqpw1 4162
 Description: A condition for equality to unit power class. (Contributed by SF, 21-Jan-2015.)
Assertion
Ref Expression
eqpw1 (A = 1B ↔ (A 1c x({x} Ax B)))
Distinct variable groups:   x,A   x,B

Proof of Theorem eqpw1
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 pw1ss1c 4158 . . 3 1B 1c
2 sseq1 3292 . . 3 (A = 1B → (A 1c1B 1c))
31, 2mpbiri 224 . 2 (A = 1BA 1c)
4 ssofeq 4077 . . . 4 ((A 1c 1B 1c) → (A = 1By 1c (y Ay 1B)))
51, 4mpan2 652 . . 3 (A 1c → (A = 1By 1c (y Ay 1B)))
6 df-ral 2619 . . . . 5 (y 1c (y Ay 1B) ↔ y(y 1c → (y Ay 1B)))
7 el1c 4139 . . . . . . . . 9 (y 1cx y = {x})
87imbi1i 315 . . . . . . . 8 ((y 1c → (y Ay 1B)) ↔ (x y = {x} → (y Ay 1B)))
9 19.23v 1891 . . . . . . . 8 (x(y = {x} → (y Ay 1B)) ↔ (x y = {x} → (y Ay 1B)))
108, 9bitr4i 243 . . . . . . 7 ((y 1c → (y Ay 1B)) ↔ x(y = {x} → (y Ay 1B)))
1110albii 1566 . . . . . 6 (y(y 1c → (y Ay 1B)) ↔ yx(y = {x} → (y Ay 1B)))
12 alcom 1737 . . . . . 6 (xy(y = {x} → (y Ay 1B)) ↔ yx(y = {x} → (y Ay 1B)))
1311, 12bitr4i 243 . . . . 5 (y(y 1c → (y Ay 1B)) ↔ xy(y = {x} → (y Ay 1B)))
146, 13bitri 240 . . . 4 (y 1c (y Ay 1B) ↔ xy(y = {x} → (y Ay 1B)))
15 snex 4111 . . . . . . 7 {x} V
16 eleq1 2413 . . . . . . . 8 (y = {x} → (y A ↔ {x} A))
17 eleq1 2413 . . . . . . . 8 (y = {x} → (y 1B ↔ {x} 1B))
1816, 17bibi12d 312 . . . . . . 7 (y = {x} → ((y Ay 1B) ↔ ({x} A ↔ {x} 1B)))
1915, 18ceqsalv 2885 . . . . . 6 (y(y = {x} → (y Ay 1B)) ↔ ({x} A ↔ {x} 1B))
20 snelpw1 4146 . . . . . . 7 ({x} 1Bx B)
2120bibi2i 304 . . . . . 6 (({x} A ↔ {x} 1B) ↔ ({x} Ax B))
2219, 21bitri 240 . . . . 5 (y(y = {x} → (y Ay 1B)) ↔ ({x} Ax B))
2322albii 1566 . . . 4 (xy(y = {x} → (y Ay 1B)) ↔ x({x} Ax B))
2414, 23bitri 240 . . 3 (y 1c (y Ay 1B) ↔ x({x} Ax B))
255, 24syl6bb 252 . 2 (A 1c → (A = 1Bx({x} Ax B)))
263, 25biadan2 623 1 (A = 1B ↔ (A 1c x({x} Ax B)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ⊆ wss 3257  {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-ral 2619  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:  eqpw1relk  4479
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