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Theorem eladdci 4399
Description: Inference form of membership in cardinal addition. (Contributed by SF, 26-Jan-2015.)
Assertion
Ref Expression
eladdci

Proof of Theorem eladdci
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2353 . . . 4
2 ineq1 3450 . . . . . . . 8
32eqeq1d 2361 . . . . . . 7
4 uneq1 3411 . . . . . . . 8
54eqeq2d 2364 . . . . . . 7
63, 5anbi12d 691 . . . . . 6
7 ineq2 3451 . . . . . . . 8
87eqeq1d 2361 . . . . . . 7
9 uneq2 3412 . . . . . . . 8
109eqeq2d 2364 . . . . . . 7
118, 10anbi12d 691 . . . . . 6
126, 11rspc2ev 2963 . . . . 5
13123expa 1151 . . . 4
141, 13mpanr2 665 . . 3
15143impa 1146 . 2
16 eladdc 4398 . 2
1715, 16sylibr 203 1
Colors of variables: wff setvar class
Syntax hints:   wi 4   wa 358   w3a 934   wceq 1642   wcel 1710  wrex 2615   cun 3207   cin 3208  c0 3550   cplc 4375
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 4078
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-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-addc 4378
This theorem is referenced by:  ncfindi  4475  tfindi  4496  nnadjoinpw  4521  sfinltfin  4535  ncdisjun  6136  tcdi  6164  ce0addcnnul  6179
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