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Theorem infxpen 10052
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 2735 . 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 2822 . . . . 5 (𝑠 = 𝑧 → (𝑠 ∈ (On × On) ↔ 𝑧 ∈ (On × On)))
3 eleq1w 2822 . . . . 5 (𝑡 = 𝑤 → (𝑡 ∈ (On × On) ↔ 𝑤 ∈ (On × On)))
42, 3bi2anan9 638 . . . 4 ((𝑠 = 𝑧𝑡 = 𝑤) → ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ↔ (𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On))))
5 fveq2 6907 . . . . . . . 8 (𝑠 = 𝑧 → (1st𝑠) = (1st𝑧))
6 fveq2 6907 . . . . . . . 8 (𝑠 = 𝑧 → (2nd𝑠) = (2nd𝑧))
75, 6uneq12d 4179 . . . . . . 7 (𝑠 = 𝑧 → ((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑧) ∪ (2nd𝑧)))
87adantr 480 . . . . . 6 ((𝑠 = 𝑧𝑡 = 𝑤) → ((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑧) ∪ (2nd𝑧)))
9 fveq2 6907 . . . . . . . 8 (𝑡 = 𝑤 → (1st𝑡) = (1st𝑤))
10 fveq2 6907 . . . . . . . 8 (𝑡 = 𝑤 → (2nd𝑡) = (2nd𝑤))
119, 10uneq12d 4179 . . . . . . 7 (𝑡 = 𝑤 → ((1st𝑡) ∪ (2nd𝑡)) = ((1st𝑤) ∪ (2nd𝑤)))
1211adantl 481 . . . . . 6 ((𝑠 = 𝑧𝑡 = 𝑤) → ((1st𝑡) ∪ (2nd𝑡)) = ((1st𝑤) ∪ (2nd𝑤)))
138, 12eleq12d 2833 . . . . 5 ((𝑠 = 𝑧𝑡 = 𝑤) → (((1st𝑠) ∪ (2nd𝑠)) ∈ ((1st𝑡) ∪ (2nd𝑡)) ↔ ((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤))))
147, 11eqeqan12d 2749 . . . . . 6 ((𝑠 = 𝑧𝑡 = 𝑤) → (((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑡) ∪ (2nd𝑡)) ↔ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
15 breq12 5153 . . . . . 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 632 . . . . 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 918 . . . 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 632 . . 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 5221 . 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 2735 . 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 261 . 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 2735 . 2 ((1st𝑤) ∪ (2nd𝑤)) = ((1st𝑤) ∪ (2nd𝑤))
23 eqid 2735 . 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 10051 1 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  wral 3059  cun 3961  cin 3962  wss 3963   class class class wbr 5148  {copab 5210   × cxp 5687  Oncon0 6386  cfv 6563  ωcom 7887  1st c1st 8011  2nd c2nd 8012  cen 8981  csdm 8983  OrdIsocoi 9547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-card 9977
This theorem is referenced by:  xpomen  10053  infxpidm2  10055  alephreg  10620  cfpwsdom  10622  inar1  10813
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