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Theorem infxpen 10054

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.

Step	Hyp	Ref	Expression
1		eqid 2737	. 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		eleq1w 2824	. . . . 5 ⊢ (𝑠 = 𝑧 → (𝑠 ∈ (On × On) ↔ 𝑧 ∈ (On × On)))
3		eleq1w 2824	. . . . 5 ⊢ (𝑡 = 𝑤 → (𝑡 ∈ (On × On) ↔ 𝑤 ∈ (On × On)))
4	2, 3	bi2anan9 638	. . . 4 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ↔ (𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On))))
5		fveq2 6906	. . . . . . . 8 ⊢ (𝑠 = 𝑧 → (1^st ‘𝑠) = (1^st ‘𝑧))
6		fveq2 6906	. . . . . . . 8 ⊢ (𝑠 = 𝑧 → (2^nd ‘𝑠) = (2^nd ‘𝑧))
7	5, 6	uneq12d 4169	. . . . . . 7 ⊢ (𝑠 = 𝑧 → ((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)))
8	7	adantr 480	. . . . . 6 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → ((1^st ‘𝑠) ∪ (2^nd ‘𝑠)) = ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)))
9		fveq2 6906	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (1^st ‘𝑡) = (1^st ‘𝑤))
10		fveq2 6906	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (2^nd ‘𝑡) = (2^nd ‘𝑤))
11	9, 10	uneq12d 4169	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)))
12	11	adantl 481	. . . . . 6 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → ((1^st ‘𝑡) ∪ (2^nd ‘𝑡)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)))
13	8, 12	eleq12d 2835	. . . . 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 2751	. . . . . 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 5148	. . . . . 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 632	. . . . 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 919	. . . 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 632	. . 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 5216	. 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 2737	. 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 261	. 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 2737	. 2 ⊢ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤))
23		eqid 2737	. 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 10053	1 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 {copab 5205 × cxp 5683 Oncon0 6384 ‘cfv 6561 ωcom 7887 1^st c1st 8012 2^nd c2nd 8013 ≈ cen 8982 ≺ csdm 8984 OrdIsocoi 9549

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-oi 9550 df-card 9979

This theorem is referenced by: xpomen 10055 infxpidm2 10057 alephreg 10622 cfpwsdom 10624 inar1 10815

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