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Theorem pw111 4170
 Description: The unit power class operation is one-to-one. (Contributed by SF, 26-Feb-2015.)
Assertion
Ref Expression
pw111 (1A = 1BA = B)

Proof of Theorem pw111
Dummy variables t x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4111 . . . . 5 {x} V
2 eleq1 2413 . . . . . 6 (t = {x} → (t 1A ↔ {x} 1A))
3 eleq1 2413 . . . . . 6 (t = {x} → (t 1B ↔ {x} 1B))
42, 3bibi12d 312 . . . . 5 (t = {x} → ((t 1At 1B) ↔ ({x} 1A ↔ {x} 1B)))
51, 4ceqsalv 2885 . . . 4 (t(t = {x} → (t 1At 1B)) ↔ ({x} 1A ↔ {x} 1B))
6 snelpw1 4146 . . . . 5 ({x} 1Ax A)
7 snelpw1 4146 . . . . 5 ({x} 1Bx B)
86, 7bibi12i 306 . . . 4 (({x} 1A ↔ {x} 1B) ↔ (x Ax B))
95, 8bitri 240 . . 3 (t(t = {x} → (t 1At 1B)) ↔ (x Ax B))
109albii 1566 . 2 (xt(t = {x} → (t 1At 1B)) ↔ x(x Ax B))
11 pw1ss1c 4158 . . . 4 1A 1c
12 pw1ss1c 4158 . . . 4 1B 1c
13 ssofeq 4077 . . . 4 ((1A 1c 1B 1c) → (1A = 1Bt 1c (t 1At 1B)))
1411, 12, 13mp2an 653 . . 3 (1A = 1Bt 1c (t 1At 1B))
15 df-ral 2619 . . . 4 (t 1c (t 1At 1B) ↔ t(t 1c → (t 1At 1B)))
16 el1c 4139 . . . . . . . 8 (t 1cx t = {x})
1716imbi1i 315 . . . . . . 7 ((t 1c → (t 1At 1B)) ↔ (x t = {x} → (t 1At 1B)))
18 19.23v 1891 . . . . . . 7 (x(t = {x} → (t 1At 1B)) ↔ (x t = {x} → (t 1At 1B)))
1917, 18bitr4i 243 . . . . . 6 ((t 1c → (t 1At 1B)) ↔ x(t = {x} → (t 1At 1B)))
2019albii 1566 . . . . 5 (t(t 1c → (t 1At 1B)) ↔ tx(t = {x} → (t 1At 1B)))
21 alcom 1737 . . . . 5 (tx(t = {x} → (t 1At 1B)) ↔ xt(t = {x} → (t 1At 1B)))
2220, 21bitri 240 . . . 4 (t(t 1c → (t 1At 1B)) ↔ xt(t = {x} → (t 1At 1B)))
2315, 22bitri 240 . . 3 (t 1c (t 1At 1B) ↔ xt(t = {x} → (t 1At 1B)))
2414, 23bitri 240 . 2 (1A = 1Bxt(t = {x} → (t 1At 1B)))
25 dfcleq 2347 . 2 (A = Bx(x Ax B))
2610, 24, 253bitr4i 268 1 (1A = 1BA = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀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:  pw1fnf1o  5855
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