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Theorem infxpen 8720
 Description: Every infinite ordinal is equinumerous to its Cartesian product. 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 2610 . 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 eleq1 2676 . . . . 5 (𝑠 = 𝑧 → (𝑠 ∈ (On × On) ↔ 𝑧 ∈ (On × On)))
3 eleq1 2676 . . . . 5 (𝑡 = 𝑤 → (𝑡 ∈ (On × On) ↔ 𝑤 ∈ (On × On)))
42, 3bi2anan9 913 . . . 4 ((𝑠 = 𝑧𝑡 = 𝑤) → ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ↔ (𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On))))
5 fveq2 6103 . . . . . . . 8 (𝑠 = 𝑧 → (1st𝑠) = (1st𝑧))
6 fveq2 6103 . . . . . . . 8 (𝑠 = 𝑧 → (2nd𝑠) = (2nd𝑧))
75, 6uneq12d 3730 . . . . . . 7 (𝑠 = 𝑧 → ((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑧) ∪ (2nd𝑧)))
87adantr 480 . . . . . 6 ((𝑠 = 𝑧𝑡 = 𝑤) → ((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑧) ∪ (2nd𝑧)))
9 fveq2 6103 . . . . . . . 8 (𝑡 = 𝑤 → (1st𝑡) = (1st𝑤))
10 fveq2 6103 . . . . . . . 8 (𝑡 = 𝑤 → (2nd𝑡) = (2nd𝑤))
119, 10uneq12d 3730 . . . . . . 7 (𝑡 = 𝑤 → ((1st𝑡) ∪ (2nd𝑡)) = ((1st𝑤) ∪ (2nd𝑤)))
1211adantl 481 . . . . . 6 ((𝑠 = 𝑧𝑡 = 𝑤) → ((1st𝑡) ∪ (2nd𝑡)) = ((1st𝑤) ∪ (2nd𝑤)))
138, 12eleq12d 2682 . . . . 5 ((𝑠 = 𝑧𝑡 = 𝑤) → (((1st𝑠) ∪ (2nd𝑠)) ∈ ((1st𝑡) ∪ (2nd𝑡)) ↔ ((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤))))
147, 11eqeqan12d 2626 . . . . . 6 ((𝑠 = 𝑧𝑡 = 𝑤) → (((1st𝑠) ∪ (2nd𝑠)) = ((1st𝑡) ∪ (2nd𝑡)) ↔ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
15 breq12 4588 . . . . . 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 743 . . . . 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 742 . . . 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 743 . . 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 4654 . 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 2610 . 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 250 . 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 2610 . 2 ((1st𝑤) ∪ (2nd𝑤)) = ((1st𝑤) ∪ (2nd𝑤))
23 eqid 2610 . 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 8719 1 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583  {copab 4642   × cxp 5036  Oncon0 5640  ‘cfv 5804  ωcom 6957  1st c1st 7057  2nd c2nd 7058   ≈ cen 7838   ≺ csdm 7840  OrdIsocoi 8297 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648 This theorem is referenced by:  xpomen  8721  infxpidm2  8723  alephreg  9283  cfpwsdom  9285  inar1  9476
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