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Theorem infxpen 9930

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 2820	. . . . 5 ⊢ (𝑠 = 𝑧 → (𝑠 ∈ (On × On) ↔ 𝑧 ∈ (On × On)))
3		eleq1w 2820	. . . . 5 ⊢ (𝑡 = 𝑤 → (𝑡 ∈ (On × On) ↔ 𝑤 ∈ (On × On)))
4	2, 3	bi2anan9 639	. . . 4 ⊢ ((𝑠 = 𝑧 ∧ 𝑡 = 𝑤) → ((𝑠 ∈ (On × On) ∧ 𝑡 ∈ (On × On)) ↔ (𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On))))
5		fveq2 6835	. . . . . . . 8 ⊢ (𝑠 = 𝑧 → (1^st ‘𝑠) = (1^st ‘𝑧))
6		fveq2 6835	. . . . . . . 8 ⊢ (𝑠 = 𝑧 → (2^nd ‘𝑠) = (2^nd ‘𝑧))
7	5, 6	uneq12d 4110	. . . . . . 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 6835	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (1^st ‘𝑡) = (1^st ‘𝑤))
10		fveq2 6835	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (2^nd ‘𝑡) = (2^nd ‘𝑤))
11	9, 10	uneq12d 4110	. . . . . . 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 2831	. . . . 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 5091	. . . . . 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 633	. . . . 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 633	. . 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 5159	. 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 9929	1 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 class class class wbr 5086 {copab 5148 × cxp 5623 Oncon0 6318 ‘cfv 6493 ωcom 7811 1^st c1st 7934 2^nd c2nd 7935 ≈ cen 8884 ≺ csdm 8886 OrdIsocoi 9418

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-oi 9419 df-card 9857

This theorem is referenced by: xpomen 9931 infxpidm2 9933 alephreg 10499 cfpwsdom 10501 inar1 10692

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