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Theorem infxpen 7880
 Description: Every infinite ordinal is equinumerous to its cross product. Proposition 10.39 of [TakeutiZaring] p. 94, whose proof we follow closely. The key idea is to show that the relation is a well-ordering of with the additional property that -initial segments of (where is a limit ordinal) are of cardinality at most . (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
infxpen

Proof of Theorem infxpen
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2430 . 2
2 eleq1 2490 . . . . 5
3 eleq1 2490 . . . . 5
42, 3bi2anan9 844 . . . 4
5 fveq2 5714 . . . . . . . 8
6 fveq2 5714 . . . . . . . 8
75, 6uneq12d 3489 . . . . . . 7
87adantr 452 . . . . . 6
9 fveq2 5714 . . . . . . . 8
10 fveq2 5714 . . . . . . . 8
119, 10uneq12d 3489 . . . . . . 7
1211adantl 453 . . . . . 6
138, 12eleq12d 2498 . . . . 5
147, 11eqeqan12d 2445 . . . . . 6
15 breq12 4204 . . . . . 6
1614, 15anbi12d 692 . . . . 5
1713, 16orbi12d 691 . . . 4
184, 17anbi12d 692 . . 3
1918cbvopabv 4264 . 2
20 eqid 2430 . 2
21 biid 228 . 2
22 eqid 2430 . 2
23 eqid 2430 . 2 OrdIso OrdIso
241, 19, 20, 21, 22, 23infxpenlem 7879 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 358   wa 359   wceq 1652   wcel 1725  wral 2692   cun 3305   cin 3306   wss 3307   class class class wbr 4199  copab 4252  con0 4568  com 4831   cxp 4862  cfv 5440  c1st 6333  c2nd 6334   cen 7092   csdm 7094  OrdIsocoi 7462 This theorem is referenced by:  xpomen  7881  infxpidm2  7882  alephreg  8441  cfpwsdom  8443  inar1  8634 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-oadd 6714  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-oi 7463  df-card 7810
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