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 Description: A condition for a unit power class to work out to an adjunction. (Contributed by SF, 26-Jan-2015.)
Hypotheses
Ref Expression
Assertion
Ref Expression
pw1eqadj (1C = (A ∪ {B}) ↔ xy(C = (x ∪ {y}) A = 1x B = {y}))
Distinct variable groups:   x,A,y   x,B,y   x,C,y

StepHypRef Expression
1 unieq 3900 . . . . 5 (1C = (A ∪ {B}) → 1C = (A ∪ {B}))
2 unipw1 4325 . . . . 5 1C = C
3 uniun 3910 . . . . 5 (A ∪ {B}) = (A{B})
41, 2, 33eqtr3g 2408 . . . 4 (1C = (A ∪ {B}) → C = (A{B}))
5 pw1eqadj.2 . . . . . . 7 B V
65unisn 3907 . . . . . 6 {B} = B
7 pw1ss1c 4158 . . . . . . . 8 1C 1c
8 ssun2 3427 . . . . . . . . . 10 {B} (A ∪ {B})
95snid 3760 . . . . . . . . . 10 B {B}
108, 9sselii 3270 . . . . . . . . 9 B (A ∪ {B})
11 eleq2 2414 . . . . . . . . 9 (1C = (A ∪ {B}) → (B 1CB (A ∪ {B})))
1210, 11mpbiri 224 . . . . . . . 8 (1C = (A ∪ {B}) → B 1C)
137, 12sseldi 3271 . . . . . . 7 (1C = (A ∪ {B}) → B 1c)
14 el1c 4139 . . . . . . . 8 (B 1cx B = {x})
15 vex 2862 . . . . . . . . . . . . 13 x V
1615unisn 3907 . . . . . . . . . . . 12 {x} = x
1716sneqi 3745 . . . . . . . . . . 11 {{x}} = {x}
1817eqcomi 2357 . . . . . . . . . 10 {x} = {{x}}
19 id 19 . . . . . . . . . 10 (B = {x} → B = {x})
20 unieq 3900 . . . . . . . . . . 11 (B = {x} → B = {x})
2120sneqd 3746 . . . . . . . . . 10 (B = {x} → {B} = {{x}})
2218, 19, 213eqtr4a 2411 . . . . . . . . 9 (B = {x} → B = {B})
2322exlimiv 1634 . . . . . . . 8 (x B = {x} → B = {B})
2414, 23sylbi 187 . . . . . . 7 (B 1cB = {B})
2513, 24syl 15 . . . . . 6 (1C = (A ∪ {B}) → B = {B})
266, 25syl5eq 2397 . . . . 5 (1C = (A ∪ {B}) → {B} = {B})
2726uneq2d 3418 . . . 4 (1C = (A ∪ {B}) → (A{B}) = (A ∪ {B}))
284, 27eqtrd 2385 . . 3 (1C = (A ∪ {B}) → C = (A ∪ {B}))
29 ssun1 3426 . . . . . 6 A (A ∪ {B})
30 sseq2 3293 . . . . . 6 (1C = (A ∪ {B}) → (A 1CA (A ∪ {B})))
3129, 30mpbiri 224 . . . . 5 (1C = (A ∪ {B}) → A 1C)
3231, 7syl6ss 3284 . . . 4 (1C = (A ∪ {B}) → A 1c)
33 eqpw1uni 4330 . . . 4 (A 1cA = 1A)
3432, 33syl 15 . . 3 (1C = (A ∪ {B}) → A = 1A)
35 pw1eqadj.1 . . . . 5 A V
3635uniex 4317 . . . 4 A V
375uniex 4317 . . . 4 B V
38 sneq 3744 . . . . . . 7 (y = B → {y} = {B})
39 uneq12 3413 . . . . . . 7 ((x = A {y} = {B}) → (x ∪ {y}) = (A ∪ {B}))
4038, 39sylan2 460 . . . . . 6 ((x = A y = B) → (x ∪ {y}) = (A ∪ {B}))
4140eqeq2d 2364 . . . . 5 ((x = A y = B) → (C = (x ∪ {y}) ↔ C = (A ∪ {B})))
42 pw1eq 4143 . . . . . . 7 (x = A1x = 1A)
4342eqeq2d 2364 . . . . . 6 (x = A → (A = 1xA = 1A))
4443adantr 451 . . . . 5 ((x = A y = B) → (A = 1xA = 1A))
4538eqeq2d 2364 . . . . . 6 (y = B → (B = {y} ↔ B = {B}))
4645adantl 452 . . . . 5 ((x = A y = B) → (B = {y} ↔ B = {B}))
4741, 44, 463anbi123d 1252 . . . 4 ((x = A y = B) → ((C = (x ∪ {y}) A = 1x B = {y}) ↔ (C = (A ∪ {B}) A = 1A B = {B})))
4836, 37, 47spc2ev 2947 . . 3 ((C = (A ∪ {B}) A = 1A B = {B}) → xy(C = (x ∪ {y}) A = 1x B = {y}))
4928, 34, 25, 48syl3anc 1182 . 2 (1C = (A ∪ {B}) → xy(C = (x ∪ {y}) A = 1x B = {y}))
50 pw1un 4163 . . . . 5 1(x ∪ {y}) = (1x1{y})
51 vex 2862 . . . . . . 7 y V
5251pw1sn 4165 . . . . . 6 1{y} = {{y}}
5352uneq2i 3415 . . . . 5 (1x1{y}) = (1x ∪ {{y}})
5450, 53eqtri 2373 . . . 4 1(x ∪ {y}) = (1x ∪ {{y}})
55 pw1eq 4143 . . . . . 6 (C = (x ∪ {y}) → 1C = 1(x ∪ {y}))
56 sneq 3744 . . . . . . 7 (B = {y} → {B} = {{y}})
57 uneq12 3413 . . . . . . 7 ((A = 1x {B} = {{y}}) → (A ∪ {B}) = (1x ∪ {{y}}))
5856, 57sylan2 460 . . . . . 6 ((A = 1x B = {y}) → (A ∪ {B}) = (1x ∪ {{y}}))
5955, 58eqeqan12d 2368 . . . . 5 ((C = (x ∪ {y}) (A = 1x B = {y})) → (1C = (A ∪ {B}) ↔ 1(x ∪ {y}) = (1x ∪ {{y}})))
60593impb 1147 . . . 4 ((C = (x ∪ {y}) A = 1x B = {y}) → (1C = (A ∪ {B}) ↔ 1(x ∪ {y}) = (1x ∪ {{y}})))
6154, 60mpbiri 224 . . 3 ((C = (x ∪ {y}) A = 1x B = {y}) → 1C = (A ∪ {B}))
6261exlimivv 1635 . 2 (xy(C = (x ∪ {y}) A = 1x B = {y}) → 1C = (A ∪ {B}))
6349, 62impbii 180 1 (1C = (A ∪ {B}) ↔ xy(C = (x ∪ {y}) A = 1x B = {y}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207   ⊆ wss 3257  {csn 3737  ∪cuni 3891  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-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-sbc 3047  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-pr 3742  df-uni 3892  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193 This theorem is referenced by:  ncfinlower  4483  sfindbl  4530
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