Theorem addcid1 4405

Description: Cardinal zero is a fixed point for cardinal addition. Theorem X.1.8 of [Rosser] p. 276. (Contributed by SF, 16-Jan-2015.)

Assertion

Ref	Expression
addcid1	|- (A +o 0c) = A

Proof of Theorem addcid1

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-0c 4377	. . 3 |- 0c = {(/)}
2	1	addceq2i 4387	. 2 |- (A +o 0c) = (A +o {(/)})
3		0ex 4110	. . . . . . 7 |- (/) e. _V
4		ineq2 3451	. . . . . . . . . 10 |- (z = (/) -> (y i^i z) = (y i^i (/)))
5	4	eqeq1d 2361	. . . . . . . . 9 |- (z = (/) -> ((y i^i z) = (/) <-> (y i^i (/)) = (/)))
6		uneq2 3412	. . . . . . . . . 10 |- (z = (/) -> (y u. z) = (y u. (/)))
7	6	eqeq2d 2364	. . . . . . . . 9 |- (z = (/) -> (x = (y u. z) <-> x = (y u. (/))))
8	5, 7	anbi12d 691	. . . . . . . 8 |- (z = (/) -> (((y i^i z) = (/) /\ x = (y u. z)) <-> ((y i^i (/)) = (/) /\ x = (y u. (/)))))
9		in0 3576	. . . . . . . . 9 |- (y i^i (/)) = (/)
10	9	biantrur 492	. . . . . . . 8 |- (x = (y u. (/)) <-> ((y i^i (/)) = (/) /\ x = (y u. (/))))
11	8, 10	syl6bbr 254	. . . . . . 7 |- (z = (/) -> (((y i^i z) = (/) /\ x = (y u. z)) <-> x = (y u. (/))))
12	3, 11	rexsn 3768	. . . . . 6 |- (E.z e. {(/)} ((y i^i z) = (/) /\ x = (y u. z)) <-> x = (y u. (/)))
13		un0 3575	. . . . . . 7 |- (y u. (/)) = y
14	13	eqeq2i 2363	. . . . . 6 |- (x = (y u. (/)) <-> x = y)
15		equcom 1680	. . . . . 6 |- (x = y <-> y = x)
16	12, 14, 15	3bitri 262	. . . . 5 |- (E.z e. {(/)} ((y i^i z) = (/) /\ x = (y u. z)) <-> y = x)
17	16	rexbii 2639	. . . 4 |- (E.y e. A E.z e. {(/)} ((y i^i z) = (/) /\ x = (y u. z)) <-> E.y e. A y = x)
18		eladdc 4398	. . . 4 |- (x e. (A +o {(/)}) <-> E.y e. A E.z e. {(/)} ((y i^i z) = (/) /\ x = (y u. z)))
19		risset 2661	. . . 4 |- (x e. A <-> E.y e. A y = x)
20	17, 18, 19	3bitr4i 268	. . 3 |- (x e. (A +o {(/)}) <-> x e. A)
21	20	eqriv 2350	. 2 |- (A +o {(/)}) = A
22	2, 21	eqtri 2373	1 |- (A +o 0c) = A

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2615   u. cun 3207   i^i cin 3208  (/)c0 3550  {csn 3737  0_cc0c 4374   +o cplc 4375

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-sik 4192  df-ssetk 4193  df-0c 4377  df-addc 4378

This theorem is referenced by:  addcid2  4407  1cnnc  4408  nncaddccl  4419  ltfinirr  4457  ltfinp1  4462  lefinlteq  4463  lefinrflx  4467  vfin1cltv  4547  nclenn  6248  ncslesuc  6266  nncdiv3  6275  nnc3n3p1  6276  nchoicelem17  6303