Theorem el2c 6192

Description: Membership in cardinal two. (Contributed by SF, 3-Mar-2015.)

Assertion

Ref	Expression
el2c	⊢ (A ∈ 2c ↔ ∃x∃y(x ≠ y ∧ A = {x, y}))

Distinct variable group:   x,A,y

Proof of Theorem el2c

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsuc 4414	. . 3 ⊢ (A ∈ (1c +c 1c) ↔ ∃t ∈ 1c ∃y ∈ ∼ tA = (t ∪ {y}))
2		df-rex 2621	. . 3 ⊢ (∃t ∈ 1c ∃y ∈ ∼ tA = (t ∪ {y}) ↔ ∃t(t ∈ 1c ∧ ∃y ∈ ∼ tA = (t ∪ {y})))
3		el1c 4140	. . . . . . 7 ⊢ (t ∈ 1c ↔ ∃x t = {x})
4	3	anbi1i 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})))
6	4, 5	bitr4i 243	. . . . 5 ⊢ ((t ∈ 1c ∧ ∃y ∈ ∼ tA = (t ∪ {y})) ↔ ∃x(t = {x} ∧ ∃y ∈ ∼ tA = (t ∪ {y})))
7	6	exbii 1582	. . . 4 ⊢ (∃t(t ∈ 1c ∧ ∃y ∈ ∼ tA = (t ∪ {y})) ↔ ∃t∃x(t = {x} ∧ ∃y ∈ ∼ tA = (t ∪ {y})))
8		excom 1741	. . . 4 ⊢ (∃t∃x(t = {x} ∧ ∃y ∈ ∼ tA = (t ∪ {y})) ↔ ∃x∃t(t = {x} ∧ ∃y ∈ ∼ tA = (t ∪ {y})))
9	7, 8	bitri 240	. . 3 ⊢ (∃t(t ∈ 1c ∧ ∃y ∈ ∼ tA = (t ∪ {y})) ↔ ∃x∃t(t = {x} ∧ ∃y ∈ ∼ tA = (t ∪ {y})))
10	1, 2, 9	3bitri 262	. 2 ⊢ (A ∈ (1c +c 1c) ↔ ∃x∃t(t = {x} ∧ ∃y ∈ ∼ tA = (t ∪ {y})))
11		1p1e2c 6156	. . 3 ⊢ (1c +c 1c) = 2c
12	11	eleq2i 2417	. 2 ⊢ (A ∈ (1c +c 1c) ↔ A ∈ 2c)
13		snex 4112	. . . . 5 ⊢ {x} ∈ V
14		compleq 3244	. . . . . 6 ⊢ (t = {x} → ∼ t = ∼ {x})
15		uneq1 3412	. . . . . . . 8 ⊢ (t = {x} → (t ∪ {y}) = ({x} ∪ {y}))
16		df-pr 3743	. . . . . . . 8 ⊢ {x, y} = ({x} ∪ {y})
17	15, 16	syl6eqr 2403	. . . . . . 7 ⊢ (t = {x} → (t ∪ {y}) = {x, y})
18	17	eqeq2d 2364	. . . . . 6 ⊢ (t = {x} → (A = (t ∪ {y}) ↔ A = {x, y}))
19	14, 18	rexeqbidv 2821	. . . . 5 ⊢ (t = {x} → (∃y ∈ ∼ tA = (t ∪ {y}) ↔ ∃y ∈ ∼ {x}A = {x, y}))
20	13, 19	ceqsexv 2895	. . . 4 ⊢ (∃t(t = {x} ∧ ∃y ∈ ∼ tA = (t ∪ {y})) ↔ ∃y ∈ ∼ {x}A = {x, y})
21		df-rex 2621	. . . 4 ⊢ (∃y ∈ ∼ {x}A = {x, y} ↔ ∃y(y ∈ ∼ {x} ∧ A = {x, y}))
22		elsn 3749	. . . . . . . . 9 ⊢ (y ∈ {x} ↔ y = x)
23		equcom 1680	. . . . . . . . 9 ⊢ (y = x ↔ x = y)
24	22, 23	bitri 240	. . . . . . . 8 ⊢ (y ∈ {x} ↔ x = y)
25	24	notbii 287	. . . . . . 7 ⊢ (¬ y ∈ {x} ↔ ¬ x = y)
26		vex 2863	. . . . . . . 8 ⊢ y ∈ V
27	26	elcompl 3226	. . . . . . 7 ⊢ (y ∈ ∼ {x} ↔ ¬ y ∈ {x})
28		df-ne 2519	. . . . . . 7 ⊢ (x ≠ y ↔ ¬ x = y)
29	25, 27, 28	3bitr4i 268	. . . . . 6 ⊢ (y ∈ ∼ {x} ↔ x ≠ y)
30	29	anbi1i 676	. . . . 5 ⊢ ((y ∈ ∼ {x} ∧ A = {x, y}) ↔ (x ≠ y ∧ A = {x, y}))
31	30	exbii 1582	. . . 4 ⊢ (∃y(y ∈ ∼ {x} ∧ A = {x, y}) ↔ ∃y(x ≠ y ∧ A = {x, y}))
32	20, 21, 31	3bitri 262	. . 3 ⊢ (∃t(t = {x} ∧ ∃y ∈ ∼ tA = (t ∪ {y})) ↔ ∃y(x ≠ y ∧ A = {x, y}))
33	32	exbii 1582	. 2 ⊢ (∃x∃t(t = {x} ∧ ∃y ∈ ∼ tA = (t ∪ {y})) ↔ ∃x∃y(x ≠ y ∧ A = {x, y}))
34	10, 12, 33	3bitr3i 266	1 ⊢ (A ∈ 2c ↔ ∃x∃y(x ≠ y ∧ A = {x, y}))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616   ∼ ccompl 3206   ∪ cun 3208  {csn 3738  {cpr 3739  1_cc1c 4135   +_c cplc 4376  2_cc2c 6095

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-nc 6102  df-2c 6105

This theorem is referenced by:  ce2  6193