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Theorem el2c 6191
 Description: Membership in cardinal two. (Contributed by SF, 3-Mar-2015.)
Assertion
Ref Expression
el2c (A 2cxy(xy A = {x, y}))
Distinct variable group:   x,A,y

Proof of Theorem el2c
Dummy variable t is distinct from all other variables.
StepHypRef Expression
1 elsuc 4413 . . 3 (A (1c +c 1c) ↔ t 1c y tA = (t ∪ {y}))
2 df-rex 2620 . . 3 (t 1c y tA = (t ∪ {y}) ↔ t(t 1c y tA = (t ∪ {y})))
3 el1c 4139 . . . . . . 7 (t 1cx t = {x})
43anbi1i 676 . . . . . 6 ((t 1c y tA = (t ∪ {y})) ↔ (x t = {x} y tA = (t ∪ {y})))
5 19.41v 1901 . . . . . 6 (x(t = {x} y tA = (t ∪ {y})) ↔ (x t = {x} y tA = (t ∪ {y})))
64, 5bitr4i 243 . . . . 5 ((t 1c y tA = (t ∪ {y})) ↔ x(t = {x} y tA = (t ∪ {y})))
76exbii 1582 . . . 4 (t(t 1c y tA = (t ∪ {y})) ↔ tx(t = {x} y tA = (t ∪ {y})))
8 excom 1741 . . . 4 (tx(t = {x} y tA = (t ∪ {y})) ↔ xt(t = {x} y tA = (t ∪ {y})))
97, 8bitri 240 . . 3 (t(t 1c y tA = (t ∪ {y})) ↔ xt(t = {x} y tA = (t ∪ {y})))
101, 2, 93bitri 262 . 2 (A (1c +c 1c) ↔ xt(t = {x} y tA = (t ∪ {y})))
11 1p1e2c 6155 . . 3 (1c +c 1c) = 2c
1211eleq2i 2417 . 2 (A (1c +c 1c) ↔ A 2c)
13 snex 4111 . . . . 5 {x} V
14 compleq 3243 . . . . . 6 (t = {x} → ∼ t = ∼ {x})
15 uneq1 3411 . . . . . . . 8 (t = {x} → (t ∪ {y}) = ({x} ∪ {y}))
16 df-pr 3742 . . . . . . . 8 {x, y} = ({x} ∪ {y})
1715, 16syl6eqr 2403 . . . . . . 7 (t = {x} → (t ∪ {y}) = {x, y})
1817eqeq2d 2364 . . . . . 6 (t = {x} → (A = (t ∪ {y}) ↔ A = {x, y}))
1914, 18rexeqbidv 2820 . . . . 5 (t = {x} → (y tA = (t ∪ {y}) ↔ y ∼ {x}A = {x, y}))
2013, 19ceqsexv 2894 . . . 4 (t(t = {x} y tA = (t ∪ {y})) ↔ y ∼ {x}A = {x, y})
21 df-rex 2620 . . . 4 (y ∼ {x}A = {x, y} ↔ y(y ∼ {x} A = {x, y}))
22 elsn 3748 . . . . . . . . 9 (y {x} ↔ y = x)
23 equcom 1680 . . . . . . . . 9 (y = xx = y)
2422, 23bitri 240 . . . . . . . 8 (y {x} ↔ x = y)
2524notbii 287 . . . . . . 7 y {x} ↔ ¬ x = y)
26 vex 2862 . . . . . . . 8 y V
2726elcompl 3225 . . . . . . 7 (y ∼ {x} ↔ ¬ y {x})
28 df-ne 2518 . . . . . . 7 (xy ↔ ¬ x = y)
2925, 27, 283bitr4i 268 . . . . . 6 (y ∼ {x} ↔ xy)
3029anbi1i 676 . . . . 5 ((y ∼ {x} A = {x, y}) ↔ (xy A = {x, y}))
3130exbii 1582 . . . 4 (y(y ∼ {x} A = {x, y}) ↔ y(xy A = {x, y}))
3220, 21, 313bitri 262 . . 3 (t(t = {x} y tA = (t ∪ {y})) ↔ y(xy A = {x, y}))
3332exbii 1582 . 2 (xt(t = {x} y tA = (t ∪ {y})) ↔ xy(xy A = {x, y}))
3410, 12, 333bitr3i 266 1 (A 2cxy(xy A = {x, y}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615   ∼ ccompl 3205   ∪ cun 3207  {csn 3737  {cpr 3738  1cc1c 4134   +c cplc 4375  2cc2c 6094 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-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101  df-2c 6104 This theorem is referenced by:  ce2  6192
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