Theorem infxpidm2 9773

Description: Every infinite well-orderable set is equinumerous to its Cartesian square. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 10318. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
infxpidm2	⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)

Proof of Theorem infxpidm2

Step	Hyp	Ref	Expression
1		cardid2 9711	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2	1	ensymd 8791	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3		xpen 8927	. . . . 5 ⊢ ((𝐴 ≈ (card‘𝐴) ∧ 𝐴 ≈ (card‘𝐴)) → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
4	2, 2, 3	syl2anc 584	. . . 4 ⊢ (𝐴 ∈ dom card → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
5	4	adantr 481	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
6		cardon 9702	. . . 4 ⊢ (card‘𝐴) ∈ On
7		cardom 9744	. . . . 5 ⊢ (card‘ω) = ω
8		omelon 9404	. . . . . . . 8 ⊢ ω ∈ On
9		onenon 9707	. . . . . . . 8 ⊢ (ω ∈ On → ω ∈ dom card)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ ω ∈ dom card
11		carddom2 9735	. . . . . . 7 ⊢ ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
12	10, 11	mpan 687	. . . . . 6 ⊢ (𝐴 ∈ dom card → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
13	12	biimpar 478	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘ω) ⊆ (card‘𝐴))
14	7, 13	eqsstrrid 3970	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ⊆ (card‘𝐴))
15		infxpen 9770	. . . 4 ⊢ (((card‘𝐴) ∈ On ∧ ω ⊆ (card‘𝐴)) → ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴))
16	6, 14, 15	sylancr 587	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴))
17		entr 8792	. . 3 ⊢ (((𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)) ∧ ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴)) → (𝐴 × 𝐴) ≈ (card‘𝐴))
18	5, 16, 17	syl2anc 584	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ (card‘𝐴))
19	1	adantr 481	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ≈ 𝐴)
20		entr 8792	. 2 ⊢ (((𝐴 × 𝐴) ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
21	18, 19, 20	syl2anc 584	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106   ⊆ wss 3887   class class class wbr 5074   × cxp 5587  dom cdm 5589  Oncon0 6266  ‘cfv 6433  ωcom 7712   ≈ cen 8730   ≼ cdom 8731  cardccrd 9693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-oi 9269  df-card 9697

This theorem is referenced by:  infpwfien  9818  mappwen  9868  infdjuabs  9962  infxpdom  9967  fin67  10151  infxpidm  10318  ttac  40858