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Theorem infxpen 9428
Description: Every infinite ordinal is equinumerous to its Cartesian square. Proposition 10.39 of [TakeutiZaring] p. 94, whose proof we follow closely. The key idea is to show that the relation 𝑅 is a well-ordering of (On × On) with the additional property that 𝑅-initial segments of (𝑥 × 𝑥) (where 𝑥 is a limit ordinal) are of cardinality at most 𝑥. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
infxpen ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)

Proof of Theorem infxpen
Dummy variables 𝑚 𝑎 𝑠 𝑡 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2818 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}
2 eleq1w 2892 . . . . 5 (𝑠 = 𝑧 → (𝑠 ∈ (On × On) ↔ 𝑧 ∈ (On × On)))
3 eleq1w 2892 . . . . 5 (𝑡 = 𝑤 → (𝑡 ∈ (On × On) ↔ 𝑤 ∈ (On × On)))
42, 3bi2anan9 635 . . . 4 ((𝑠 = 𝑧𝑡 = 𝑤) → ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ↔ (𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On))))
5 fveq2 6663 . . . . . . . 8 (𝑠 = 𝑧 → (1st𝑠) = (1st𝑧))
6 fveq2 6663 . . . . . . . 8 (𝑠 = 𝑧 → (2nd𝑠) = (2nd𝑧))
75, 6uneq12d 4137 . . . . . . 7 (𝑠 = 𝑧 → ((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑧) ∪ (2nd𝑧)))
87adantr 481 . . . . . 6 ((𝑠 = 𝑧𝑡 = 𝑤) → ((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑧) ∪ (2nd𝑧)))
9 fveq2 6663 . . . . . . . 8 (𝑡 = 𝑤 → (1st𝑡) = (1st𝑤))
10 fveq2 6663 . . . . . . . 8 (𝑡 = 𝑤 → (2nd𝑡) = (2nd𝑤))
119, 10uneq12d 4137 . . . . . . 7 (𝑡 = 𝑤 → ((1st𝑡) ∪ (2nd𝑡)) = ((1st𝑤) ∪ (2nd𝑤)))
1211adantl 482 . . . . . 6 ((𝑠 = 𝑧𝑡 = 𝑤) → ((1st𝑡) ∪ (2nd𝑡)) = ((1st𝑤) ∪ (2nd𝑤)))
138, 12eleq12d 2904 . . . . 5 ((𝑠 = 𝑧𝑡 = 𝑤) → (((1st𝑠) ∪ (2nd𝑠)) ∈ ((1st𝑡) ∪ (2nd𝑡)) ↔ ((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤))))
147, 11eqeqan12d 2835 . . . . . 6 ((𝑠 = 𝑧𝑡 = 𝑤) → (((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑡) ∪ (2nd𝑡)) ↔ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
15 breq12 5062 . . . . . 6 ((𝑠 = 𝑧𝑡 = 𝑤) → (𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑡𝑧{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑤))
1614, 15anbi12d 630 . . . . 5 ((𝑠 = 𝑧𝑡 = 𝑤) → ((((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑡) ∪ (2nd𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑡) ↔ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑤)))
1713, 16orbi12d 912 . . . 4 ((𝑠 = 𝑧𝑡 = 𝑤) → ((((1st𝑠) ∪ (2nd𝑠)) ∈ ((1st𝑡) ∪ (2nd𝑡)) ∨ (((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑡) ∪ (2nd𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑡)) ↔ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑤))))
184, 17anbi12d 630 . . 3 ((𝑠 = 𝑧𝑡 = 𝑤) → (((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ∧ (((1st𝑠) ∪ (2nd𝑠)) ∈ ((1st𝑡) ∪ (2nd𝑡)) ∨ (((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑡) ∪ (2nd𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑡))) ↔ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑤)))))
1918cbvopabv 5129 . 2 {⟨𝑠, 𝑡⟩ ∣ ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ∧ (((1st𝑠) ∪ (2nd𝑠)) ∈ ((1st𝑡) ∪ (2nd𝑡)) ∨ (((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑡) ∪ (2nd𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑡)))} = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑤)))}
20 eqid 2818 . 2 ({⟨𝑠, 𝑡⟩ ∣ ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ∧ (((1st𝑠) ∪ (2nd𝑠)) ∈ ((1st𝑡) ∪ (2nd𝑡)) ∨ (((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑡) ∪ (2nd𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑡)))} ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) = ({⟨𝑠, 𝑡⟩ ∣ ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ∧ (((1st𝑠) ∪ (2nd𝑠)) ∈ ((1st𝑡) ∪ (2nd𝑡)) ∨ (((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑡) ∪ (2nd𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑡)))} ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))
21 biid 262 . 2 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) ↔ ((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)))
22 eqid 2818 . 2 ((1st𝑤) ∪ (2nd𝑤)) = ((1st𝑤) ∪ (2nd𝑤))
23 eqid 2818 . 2 OrdIso(({⟨𝑠, 𝑡⟩ ∣ ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ∧ (((1st𝑠) ∪ (2nd𝑠)) ∈ ((1st𝑡) ∪ (2nd𝑡)) ∨ (((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑡) ∪ (2nd𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑡)))} ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))), (𝑎 × 𝑎)) = OrdIso(({⟨𝑠, 𝑡⟩ ∣ ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ∧ (((1st𝑠) ∪ (2nd𝑠)) ∈ ((1st𝑡) ∪ (2nd𝑡)) ∨ (((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑡) ∪ (2nd𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}𝑡)))} ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))), (𝑎 × 𝑎))
241, 19, 20, 21, 22, 23infxpenlem 9427 1 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  wral 3135  cun 3931  cin 3932  wss 3933   class class class wbr 5057  {copab 5119   × cxp 5546  Oncon0 6184  cfv 6348  ωcom 7569  1st c1st 7676  2nd c2nd 7677  cen 8494  csdm 8496  OrdIsocoi 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-card 9356
This theorem is referenced by:  xpomen  9429  infxpidm2  9431  alephreg  9992  cfpwsdom  9994  inar1  10185
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