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Theorem clos1nrel 5886
 Description: The value of a closure when the base set is not related to anything in R. (Contributed by SF, 13-Mar-2015.)
Hypotheses
Ref Expression
clos1nrel.1 S V
clos1nrel.2 R V
clos1nrel.3 C = Clos1 (S, R)
Assertion
Ref Expression
clos1nrel ((RS) = C = S)

Proof of Theorem clos1nrel
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eq0 3564 . . . . 5 ((RS) = y ¬ y (RS))
2 rspe 2675 . . . . . . . . 9 ((x S xRy) → x S xRy)
3 elima 4754 . . . . . . . . 9 (y (RS) ↔ x S xRy)
42, 3sylibr 203 . . . . . . . 8 ((x S xRy) → y (RS))
54con3i 127 . . . . . . 7 y (RS) → ¬ (x S xRy))
65pm2.21d 98 . . . . . 6 y (RS) → ((x S xRy) → y S))
76alimi 1559 . . . . 5 (y ¬ y (RS) → y((x S xRy) → y S))
81, 7sylbi 187 . . . 4 ((RS) = y((x S xRy) → y S))
98ralrimivw 2698 . . 3 ((RS) = x C y((x S xRy) → y S))
10 clos1nrel.1 . . . 4 S V
11 ssid 3290 . . . 4 S S
12 clos1nrel.2 . . . . 5 R V
13 clos1nrel.3 . . . . 5 C = Clos1 (S, R)
1410, 12, 13clos1induct 5880 . . . 4 ((S V S S x C y((x S xRy) → y S)) → C S)
1510, 11, 14mp3an12 1267 . . 3 (x C y((x S xRy) → y S) → C S)
169, 15syl 15 . 2 ((RS) = C S)
1713clos1base 5878 . . 3 S C
1817a1i 10 . 2 ((RS) = S C)
1916, 18eqssd 3289 1 ((RS) = C = S)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  ∅c0 3550   class class class wbr 4639   “ cima 4722   Clos1 cclos1 5872 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-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-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-2nd 4797  df-txp 5736  df-fix 5740  df-ins2 5750  df-ins3 5752  df-image 5754  df-clos1 5873 This theorem is referenced by:  nchoicelem3  6291
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