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Theorem pw1disj 4167
 Description: Two unit power classes are disjoint iff the classes themselves are disjoint. (Contributed by SF, 26-Jan-2015.)
Assertion
Ref Expression
pw1disj ((1A1B) = ↔ (AB) = )

Proof of Theorem pw1disj
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 disj 3591 . . . . . 6 ((1A1B) = y 1 A ¬ y 1B)
2 eleq1 2413 . . . . . . . 8 (y = {x} → (y 1B ↔ {x} 1B))
32notbid 285 . . . . . . 7 (y = {x} → (¬ y 1B ↔ ¬ {x} 1B))
43rspccv 2952 . . . . . 6 (y 1 A ¬ y 1B → ({x} 1A → ¬ {x} 1B))
51, 4sylbi 187 . . . . 5 ((1A1B) = → ({x} 1A → ¬ {x} 1B))
6 snelpw1 4146 . . . . 5 ({x} 1Ax A)
7 snelpw1 4146 . . . . . 6 ({x} 1Bx B)
87notbii 287 . . . . 5 (¬ {x} 1B ↔ ¬ x B)
95, 6, 83imtr3g 260 . . . 4 ((1A1B) = → (x A → ¬ x B))
109ralrimiv 2696 . . 3 ((1A1B) = x A ¬ x B)
11 disj 3591 . . 3 ((AB) = x A ¬ x B)
1210, 11sylibr 203 . 2 ((1A1B) = → (AB) = )
13 elpw1 4144 . . . . 5 (x 1Ay A x = {y})
14 disj 3591 . . . . . . . . 9 ((AB) = y A ¬ y B)
15 rsp 2674 . . . . . . . . 9 (y A ¬ y B → (y A → ¬ y B))
1614, 15sylbi 187 . . . . . . . 8 ((AB) = → (y A → ¬ y B))
1716imp 418 . . . . . . 7 (((AB) = y A) → ¬ y B)
18 eleq1 2413 . . . . . . . . 9 (x = {y} → (x 1B ↔ {y} 1B))
19 snelpw1 4146 . . . . . . . . 9 ({y} 1By B)
2018, 19syl6bb 252 . . . . . . . 8 (x = {y} → (x 1By B))
2120notbid 285 . . . . . . 7 (x = {y} → (¬ x 1B ↔ ¬ y B))
2217, 21syl5ibrcom 213 . . . . . 6 (((AB) = y A) → (x = {y} → ¬ x 1B))
2322rexlimdva 2738 . . . . 5 ((AB) = → (y A x = {y} → ¬ x 1B))
2413, 23syl5bi 208 . . . 4 ((AB) = → (x 1A → ¬ x 1B))
2524ralrimiv 2696 . . 3 ((AB) = x 1 A ¬ x 1B)
26 disj 3591 . . 3 ((1A1B) = x 1 A ¬ x 1B)
2725, 26sylibr 203 . 2 ((AB) = → (1A1B) = )
2812, 27impbii 180 1 ((1A1B) = ↔ (AB) = )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615   ∩ cin 3208  ∅c0 3550  {csn 3737  ℘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: (None)
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