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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.

Step	Hyp	Ref	Expression
1		eqid 2610	. 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 2676	. . . . 5 ⊢ (𝑠 = 𝑧 → (𝑠 ∈ (On × On) ↔ 𝑧 ∈ (On × On)))
3		eleq1 2676	. . . . 5 ⊢ (𝑡 = 𝑤 → (𝑡 ∈ (On × On) ↔ 𝑤 ∈ (On × On)))
4	2, 3	bi2anan9 913	. . . 4 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ↔ (𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On))))
5		fveq2 6103	. . . . . . . 8 ⊢ (𝑠 = 𝑧 → (1^st ‘𝑠) = (1^st ‘𝑧))
6		fveq2 6103	. . . . . . . 8 ⊢ (𝑠 = 𝑧 → (2^nd ‘𝑠) = (2^nd ‘𝑧))
7	5, 6	uneq12d 3730	. . . . . . 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 6103	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (1^st ‘𝑡) = (1^st ‘𝑤))
10		fveq2 6103	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (2^nd ‘𝑡) = (2^nd ‘𝑤))
11	9, 10	uneq12d 3730	. . . . . . 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 2682	. . . . 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 2626	. . . . . 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 4588	. . . . . 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 743	. . . . 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 742	. . . 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 743	. . 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 4654	. 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 2610	. 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 250	. 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 2610	. 2 ⊢ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤))
23		eqid 2610	. 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 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 1^st c1st 7057 2^nd 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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