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Theorem elimapw13 4946
 Description: Membership in an image under three unit power classes. (Contributed by set.mm contributors, 18-Mar-2015.)
Assertion
Ref Expression
elimapw13 (A (B111C) ↔ x C {{{x}}}, A B)
Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem elimapw13
Dummy variable t is distinct from all other variables.
StepHypRef Expression
1 elimapw12 4945 . 2 (A (B111C) ↔ t 1 C{{t}}, A B)
2 df-rex 2620 . . 3 (t 1 C{{t}}, A Bt(t 1C {{t}}, A B))
3 elpw1 4144 . . . . . . . 8 (t 1Cx C t = {x})
43anbi1i 676 . . . . . . 7 ((t 1C {{t}}, A B) ↔ (x C t = {x} {{t}}, A B))
5 r19.41v 2764 . . . . . . 7 (x C (t = {x} {{t}}, A B) ↔ (x C t = {x} {{t}}, A B))
64, 5bitr4i 243 . . . . . 6 ((t 1C {{t}}, A B) ↔ x C (t = {x} {{t}}, A B))
76exbii 1582 . . . . 5 (t(t 1C {{t}}, A B) ↔ tx C (t = {x} {{t}}, A B))
8 rexcom4 2878 . . . . 5 (x C t(t = {x} {{t}}, A B) ↔ tx C (t = {x} {{t}}, A B))
97, 8bitr4i 243 . . . 4 (t(t 1C {{t}}, A B) ↔ x C t(t = {x} {{t}}, A B))
10 snex 4111 . . . . . 6 {x} V
11 sneq 3744 . . . . . . . . 9 (t = {x} → {t} = {{x}})
1211sneqd 3746 . . . . . . . 8 (t = {x} → {{t}} = {{{x}}})
1312opeq1d 4584 . . . . . . 7 (t = {x} → {{t}}, A = {{{x}}}, A)
1413eleq1d 2419 . . . . . 6 (t = {x} → ({{t}}, A B{{{x}}}, A B))
1510, 14ceqsexv 2894 . . . . 5 (t(t = {x} {{t}}, A B) ↔ {{{x}}}, A B)
1615rexbii 2639 . . . 4 (x C t(t = {x} {{t}}, A B) ↔ x C {{{x}}}, A B)
179, 16bitri 240 . . 3 (t(t 1C {{t}}, A B) ↔ x C {{{x}}}, A B)
182, 17bitri 240 . 2 (t 1 C{{t}}, A Bx C {{{x}}}, A B)
191, 18bitri 240 1 (A (B111C) ↔ x C {{{x}}}, A B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  {csn 3737  ℘1cpw1 4135  ⟨cop 4561   “ cima 4722 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-br 4640  df-ima 4727 This theorem is referenced by:  nchoicelem11  6299  nchoicelem16  6304
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