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 Description: Equivalence law for adjunction. Theorem XI.1.13 of [Rosser] p. 348. (Contributed by SF, 25-Feb-2015.)
Hypotheses
Ref Expression
Assertion
Ref Expression
enadj (((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) → AB)

StepHypRef Expression
1 sneq 3744 . . . . . 6 (X = Y → {X} = {Y})
21uneq2d 3418 . . . . 5 (X = Y → (A ∪ {X}) = (A ∪ {Y}))
32eqeq1d 2361 . . . 4 (X = Y → ((A ∪ {X}) = (B ∪ {Y}) ↔ (A ∪ {Y}) = (B ∪ {Y})))
4 eleq1 2413 . . . . 5 (X = Y → (X AY A))
54notbid 285 . . . 4 (X = Y → (¬ X A ↔ ¬ Y A))
63, 53anbi12d 1253 . . 3 (X = Y → (((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) ↔ ((A ∪ {Y}) = (B ∪ {Y}) ¬ Y A ¬ Y B)))
7 simp1 955 . . . . . 6 (((A ∪ {Y}) = (B ∪ {Y}) ¬ Y A ¬ Y B) → (A ∪ {Y}) = (B ∪ {Y}))
87difeq1d 3384 . . . . 5 (((A ∪ {Y}) = (B ∪ {Y}) ¬ Y A ¬ Y B) → ((A ∪ {Y}) {Y}) = ((B ∪ {Y}) {Y}))
9 nnsucelrlem2 4425 . . . . . 6 Y A → ((A ∪ {Y}) {Y}) = A)
1093ad2ant2 977 . . . . 5 (((A ∪ {Y}) = (B ∪ {Y}) ¬ Y A ¬ Y B) → ((A ∪ {Y}) {Y}) = A)
11 nnsucelrlem2 4425 . . . . . 6 Y B → ((B ∪ {Y}) {Y}) = B)
12113ad2ant3 978 . . . . 5 (((A ∪ {Y}) = (B ∪ {Y}) ¬ Y A ¬ Y B) → ((B ∪ {Y}) {Y}) = B)
138, 10, 123eqtr3d 2393 . . . 4 (((A ∪ {Y}) = (B ∪ {Y}) ¬ Y A ¬ Y B) → A = B)
14 enadj.2 . . . . 5 B V
1514enrflx 6035 . . . 4 BB
1613, 15syl6eqbr 4676 . . 3 (((A ∪ {Y}) = (B ∪ {Y}) ¬ Y A ¬ Y B) → AB)
176, 16syl6bi 219 . 2 (X = Y → (((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) → AB))
18 elsni 3757 . . . . . . . . 9 (Y {X} → Y = X)
1918eqcomd 2358 . . . . . . . 8 (Y {X} → X = Y)
2019necon3ai 2556 . . . . . . 7 (XY → ¬ Y {X})
2120adantr 451 . . . . . 6 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B)) → ¬ Y {X})
22 ssun2 3427 . . . . . . . . 9 {Y} (B ∪ {Y})
23 enadj.4 . . . . . . . . . 10 Y V
2423snid 3760 . . . . . . . . 9 Y {Y}
2522, 24sselii 3270 . . . . . . . 8 Y (B ∪ {Y})
26 simpr1 961 . . . . . . . 8 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B)) → (A ∪ {X}) = (B ∪ {Y}))
2725, 26syl5eleqr 2440 . . . . . . 7 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B)) → Y (A ∪ {X}))
28 elun 3220 . . . . . . 7 (Y (A ∪ {X}) ↔ (Y A Y {X}))
2927, 28sylib 188 . . . . . 6 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B)) → (Y A Y {X}))
30 orel2 372 . . . . . 6 Y {X} → ((Y A Y {X}) → Y A))
3121, 29, 30sylc 56 . . . . 5 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B)) → Y A)
32 elsni 3757 . . . . . . . 8 (X {Y} → X = Y)
3332necon3ai 2556 . . . . . . 7 (XY → ¬ X {Y})
3433adantr 451 . . . . . 6 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B)) → ¬ X {Y})
35 ssun2 3427 . . . . . . . . 9 {X} (A ∪ {X})
36 enadj.3 . . . . . . . . . 10 X V
3736snid 3760 . . . . . . . . 9 X {X}
3835, 37sselii 3270 . . . . . . . 8 X (A ∪ {X})
3938, 26syl5eleq 2439 . . . . . . 7 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B)) → X (B ∪ {Y}))
40 elun 3220 . . . . . . 7 (X (B ∪ {Y}) ↔ (X B X {Y}))
4139, 40sylib 188 . . . . . 6 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B)) → (X B X {Y}))
42 orel2 372 . . . . . 6 X {Y} → ((X B X {Y}) → X B))
4334, 41, 42sylc 56 . . . . 5 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B)) → X B)
4431, 43jca 518 . . . 4 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B)) → (Y A X B))
45 simpl1 958 . . . . . . 7 ((((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) → (A ∪ {X}) = (B ∪ {Y}))
46 simpl2 959 . . . . . . 7 ((((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) → ¬ X A)
47 simpl3 960 . . . . . . 7 ((((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) → ¬ Y B)
48 simprl 732 . . . . . . 7 ((((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) → Y A)
49 simprr 733 . . . . . . 7 ((((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) → X B)
50 enadjlem1 6059 . . . . . . 7 (((A ∪ {X}) = (B ∪ {Y}) X A ¬ Y B) (Y A X B)) → (A {Y}) = (B {X}))
5145, 46, 47, 48, 49, 50syl122anc 1191 . . . . . 6 ((((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) → (A {Y}) = (B {X}))
52513adant1 973 . . . . 5 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) → (A {Y}) = (B {X}))
53 enadj.1 . . . . . . . . . . 11 A V
54 snex 4111 . . . . . . . . . . 11 {Y} V
5553, 54difex 4107 . . . . . . . . . 10 (A {Y}) V
5655enrflx 6035 . . . . . . . . 9 (A {Y}) ≈ (A {Y})
57 breq2 4643 . . . . . . . . 9 ((A {Y}) = (B {X}) → ((A {Y}) ≈ (A {Y}) ↔ (A {Y}) ≈ (B {X})))
5856, 57mpbii 202 . . . . . . . 8 ((A {Y}) = (B {X}) → (A {Y}) ≈ (B {X}))
5958adantl 452 . . . . . . 7 (((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) (A {Y}) = (B {X})) → (A {Y}) ≈ (B {X}))
6023, 36ensn 6058 . . . . . . . 8 {Y} ≈ {X}
6160a1i 10 . . . . . . 7 (((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) (A {Y}) = (B {X})) → {Y} ≈ {X})
62 incom 3448 . . . . . . . . 9 ((A {Y}) ∩ {Y}) = ({Y} ∩ (A {Y}))
63 disjdif 3622 . . . . . . . . 9 ({Y} ∩ (A {Y})) =
6462, 63eqtri 2373 . . . . . . . 8 ((A {Y}) ∩ {Y}) =
6564a1i 10 . . . . . . 7 (((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) (A {Y}) = (B {X})) → ((A {Y}) ∩ {Y}) = )
66 incom 3448 . . . . . . . . 9 ((B {X}) ∩ {X}) = ({X} ∩ (B {X}))
67 disjdif 3622 . . . . . . . . 9 ({X} ∩ (B {X})) =
6866, 67eqtri 2373 . . . . . . . 8 ((B {X}) ∩ {X}) =
6968a1i 10 . . . . . . 7 (((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) (A {Y}) = (B {X})) → ((B {X}) ∩ {X}) = )
70 unen 6048 . . . . . . 7 ((((A {Y}) ≈ (B {X}) {Y} ≈ {X}) (((A {Y}) ∩ {Y}) = ((B {X}) ∩ {X}) = )) → ((A {Y}) ∪ {Y}) ≈ ((B {X}) ∪ {X}))
7159, 61, 65, 69, 70syl22anc 1183 . . . . . 6 (((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) (A {Y}) = (B {X})) → ((A {Y}) ∪ {Y}) ≈ ((B {X}) ∪ {X}))
72 simpl3l 1010 . . . . . . 7 (((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) (A {Y}) = (B {X})) → Y A)
73 nnsucelrlem4 4427 . . . . . . 7 (Y A → ((A {Y}) ∪ {Y}) = A)
7472, 73syl 15 . . . . . 6 (((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) (A {Y}) = (B {X})) → ((A {Y}) ∪ {Y}) = A)
75 simpl3r 1011 . . . . . . 7 (((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) (A {Y}) = (B {X})) → X B)
76 nnsucelrlem4 4427 . . . . . . 7 (X B → ((B {X}) ∪ {X}) = B)
7775, 76syl 15 . . . . . 6 (((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) (A {Y}) = (B {X})) → ((B {X}) ∪ {X}) = B)
7871, 74, 773brtr3d 4668 . . . . 5 (((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) (A {Y}) = (B {X})) → AB)
7952, 78mpdan 649 . . . 4 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) (Y A X B)) → AB)
8044, 79mpd3an3 1278 . . 3 ((XY ((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B)) → AB)
8180ex 423 . 2 (XY → (((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) → AB))
8217, 81pm2.61ine 2592 1 (((A ∪ {X}) = (B ∪ {Y}) ¬ X A ¬ Y B) → AB)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  Vcvv 2859   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737   class class class wbr 4639   ≈ cen 6028 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-13 1712  ax-14 1714  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-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-en 6029 This theorem is referenced by:  peano4nc  6150
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