Theorem addcid1 4406

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 +c 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 4378	. . 3 ⊢ 0c = {∅}
2	1	addceq2i 4388	. 2 ⊢ (A +c 0c) = (A +c {∅})
3		0ex 4111	. . . . . . 7 ⊢ ∅ ∈ V
4		ineq2 3452	. . . . . . . . . 10 ⊢ (z = ∅ → (y ∩ z) = (y ∩ ∅))
5	4	eqeq1d 2361	. . . . . . . . 9 ⊢ (z = ∅ → ((y ∩ z) = ∅ ↔ (y ∩ ∅) = ∅))
6		uneq2 3413	. . . . . . . . . 10 ⊢ (z = ∅ → (y ∪ z) = (y ∪ ∅))
7	6	eqeq2d 2364	. . . . . . . . 9 ⊢ (z = ∅ → (x = (y ∪ z) ↔ x = (y ∪ ∅)))
8	5, 7	anbi12d 691	. . . . . . . 8 ⊢ (z = ∅ → (((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ ((y ∩ ∅) = ∅ ∧ x = (y ∪ ∅))))
9		in0 3577	. . . . . . . . 9 ⊢ (y ∩ ∅) = ∅
10	9	biantrur 492	. . . . . . . 8 ⊢ (x = (y ∪ ∅) ↔ ((y ∩ ∅) = ∅ ∧ x = (y ∪ ∅)))
11	8, 10	syl6bbr 254	. . . . . . 7 ⊢ (z = ∅ → (((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ x = (y ∪ ∅)))
12	3, 11	rexsn 3769	. . . . . 6 ⊢ (∃z ∈ {∅} ((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ x = (y ∪ ∅))
13		un0 3576	. . . . . . 7 ⊢ (y ∪ ∅) = y
14	13	eqeq2i 2363	. . . . . 6 ⊢ (x = (y ∪ ∅) ↔ x = y)
15		equcom 1680	. . . . . 6 ⊢ (x = y ↔ y = x)
16	12, 14, 15	3bitri 262	. . . . 5 ⊢ (∃z ∈ {∅} ((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ y = x)
17	16	rexbii 2640	. . . 4 ⊢ (∃y ∈ A ∃z ∈ {∅} ((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ ∃y ∈ A y = x)
18		eladdc 4399	. . . 4 ⊢ (x ∈ (A +c {∅}) ↔ ∃y ∈ A ∃z ∈ {∅} ((y ∩ z) = ∅ ∧ x = (y ∪ z)))
19		risset 2662	. . . 4 ⊢ (x ∈ A ↔ ∃y ∈ A y = x)
20	17, 18, 19	3bitr4i 268	. . 3 ⊢ (x ∈ (A +c {∅}) ↔ x ∈ A)
21	20	eqriv 2350	. 2 ⊢ (A +c {∅}) = A
22	2, 21	eqtri 2373	1 ⊢ (A +c 0c) = A

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738  0_cc0c 4375   +_c cplc 4376

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  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-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-opk 4059  df-1c 4137  df-pw1 4138  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-sik 4193  df-ssetk 4194  df-0c 4378  df-addc 4379

This theorem is referenced by:  addcid2  4408  1cnnc  4409  nncaddccl  4420  ltfinirr  4458  ltfinp1  4463  lefinlteq  4464  lefinrflx  4468  vfin1cltv  4548  nclenn  6250  ncslesuc  6268  nncdiv3  6278  nnc3n3p1  6279  nchoicelem17  6306