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Theorem infxpen 8837

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.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}
2		eleq1 2689	. . . . 5 ⊢ (𝑠 = 𝑧 → (𝑠 ∈ (On × On) ↔ 𝑧 ∈ (On × On)))
3		eleq1 2689	. . . . 5 ⊢ (𝑡 = 𝑤 → (𝑡 ∈ (On × On) ↔ 𝑤 ∈ (On × On)))
4	2, 3	bi2anan9 917	. . . 4 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ↔ (𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On))))
5		fveq2 6191	. . . . . . . 8 ⊢ (𝑠 = 𝑧 → (1^st ‘𝑠) = (1^st ‘𝑧))
6		fveq2 6191	. . . . . . . 8 ⊢ (𝑠 = 𝑧 → (2^nd ‘𝑠) = (2^nd ‘𝑧))
7	5, 6	uneq12d 3768	. . . . . . 7 ⊢ (𝑠 = 𝑧 → ((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)))
8	7	adantr 481	. . . . . 6 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → ((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)))
9		fveq2 6191	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (1^st ‘𝑡) = (1^st ‘𝑤))
10		fveq2 6191	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (2^nd ‘𝑡) = (2^nd ‘𝑤))
11	9, 10	uneq12d 3768	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)))
12	11	adantl 482	. . . . . 6 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)))
13	8, 12	eleq12d 2695	. . . . 5 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) ∈ ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ↔ ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤))))
14	7, 11	eqeqan12d 2638	. . . . . 6 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ↔ ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤))))
15		breq12 4658	. . . . . 6 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → (𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑡 ↔ 𝑧{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑤))
16	14, 15	anbi12d 747	. . . . 5 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → ((((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑡) ↔ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∧ 𝑧{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑤)))
17	13, 16	orbi12d 746	. . . 4 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → ((((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) ∈ ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∨ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑡)) ↔ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∨ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∧ 𝑧{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑤))))
18	4, 17	anbi12d 747	. . 3 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → (((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ∧ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) ∈ ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∨ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑡))) ↔ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∨ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∧ 𝑧{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑤)))))
19	18	cbvopabv 4722	. 2 ⊢ {⟨𝑠, 𝑡⟩ ∣ ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ∧ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) ∈ ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∨ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑡)))} = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∨ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∧ 𝑧{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑤)))}
20		eqid 2622	. 2 ⊢ ({⟨𝑠, 𝑡⟩ ∣ ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ∧ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) ∈ ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∨ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑡)))} ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) = ({⟨𝑠, 𝑡⟩ ∣ ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ∧ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) ∈ ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∨ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑡)))} ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))
21		biid 251	. 2 ⊢ (((𝑎 ∈ On ∧ ∀𝑚 ∈ 𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚 ∈ 𝑎 𝑚 ≺ 𝑎)) ↔ ((𝑎 ∈ On ∧ ∀𝑚 ∈ 𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚 ∈ 𝑎 𝑚 ≺ 𝑎)))
22		eqid 2622	. 2 ⊢ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤))
23		eqid 2622	. 2 ⊢ OrdIso(({⟨𝑠, 𝑡⟩ ∣ ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ∧ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) ∈ ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∨ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑡)))} ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))), (𝑎 × 𝑎)) = OrdIso(({⟨𝑠, 𝑡⟩ ∣ ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ∧ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) ∈ ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∨ (((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) ∧ 𝑠{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}𝑡)))} ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))), (𝑎 × 𝑎))
24	1, 19, 20, 21, 22, 23	infxpenlem 8836	1 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 {copab 4712 × cxp 5112 Oncon0 5723 ‘cfv 5888 ωcom 7065 1^st c1st 7166 2^nd c2nd 7167 ≈ cen 7952 ≺ csdm 7954 OrdIsocoi 8414

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-oi 8415 df-card 8765

This theorem is referenced by: xpomen 8838 infxpidm2 8840 alephreg 9404 cfpwsdom 9406 inar1 9597

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