Theorem infxpidm 10462

Description: Every infinite class is equinumerous to its Cartesian square. This theorem, which is equivalent to the axiom of choice over ZF, provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. This is a corollary of infxpen 9914 (used via infxpidm2 9917). (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)

Assertion

Ref	Expression
infxpidm	⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴)

Proof of Theorem infxpidm

Step	Hyp	Ref	Expression
1		reldom 8883	. . . 4 ⊢ Rel ≼
2	1	brrelex2i 5678	. . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
3		numth3 10370	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ dom card)
4	2, 3	syl 17	. 2 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ dom card)
5		infxpidm2 9917	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
6	4, 5	mpancom 688	1 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2113  Vcvv 3437   class class class wbr 5095   × cxp 5619  dom cdm 5621  ωcom 7804   ≈ cen 8874   ≼ cdom 8875  cardccrd 9837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676  ax-inf2 9540  ax-ac2 10363

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-riota 7311  df-ov 7357  df-om 7805  df-1st 7929  df-2nd 7930  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-rdg 8337  df-1o 8393  df-er 8630  df-en 8878  df-dom 8879  df-sdom 8880  df-fin 8881  df-oi 9405  df-card 9841  df-ac 10016

This theorem is referenced by:  unirnfdomd  10467  inar1  10675