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 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

Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-0c 4377 . . 3 0c = {}
21addceq2i 4387 . 2 (A +c 0c) = (A +c {})
3 0ex 4110 . . . . . . 7 V
4 ineq2 3451 . . . . . . . . . 10 (z = → (yz) = (y))
54eqeq1d 2361 . . . . . . . . 9 (z = → ((yz) = ↔ (y) = ))
6 uneq2 3412 . . . . . . . . . 10 (z = → (yz) = (y))
76eqeq2d 2364 . . . . . . . . 9 (z = → (x = (yz) ↔ x = (y)))
85, 7anbi12d 691 . . . . . . . 8 (z = → (((yz) = x = (yz)) ↔ ((y) = x = (y))))
9 in0 3576 . . . . . . . . 9 (y) =
109biantrur 492 . . . . . . . 8 (x = (y) ↔ ((y) = x = (y)))
118, 10syl6bbr 254 . . . . . . 7 (z = → (((yz) = x = (yz)) ↔ x = (y)))
123, 11rexsn 3768 . . . . . 6 (z {} ((yz) = x = (yz)) ↔ x = (y))
13 un0 3575 . . . . . . 7 (y) = y
1413eqeq2i 2363 . . . . . 6 (x = (y) ↔ x = y)
15 equcom 1680 . . . . . 6 (x = yy = x)
1612, 14, 153bitri 262 . . . . 5 (z {} ((yz) = x = (yz)) ↔ y = x)
1716rexbii 2639 . . . 4 (y A z {} ((yz) = x = (yz)) ↔ y A y = x)
18 eladdc 4398 . . . 4 (x (A +c {}) ↔ y A z {} ((yz) = x = (yz)))
19 risset 2661 . . . 4 (x Ay A y = x)
2017, 18, 193bitr4i 268 . . 3 (x (A +c {}) ↔ x A)
2120eqriv 2350 . 2 (A +c {}) = A
222, 21eqtri 2373 1 (A +c 0c) = A
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  0cc0c 4374   +c 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  6249  ncslesuc  6267  nncdiv3  6277  nnc3n3p1  6278  nchoicelem17  6305
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