|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > infxpidm | Structured version Visualization version GIF version | ||
| 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 10054 (used via infxpidm2 10057). (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.) | 
| Ref | Expression | 
|---|---|
| infxpidm | ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | reldom 8991 | . . . 4 ⊢ Rel ≼ | |
| 2 | 1 | brrelex2i 5742 | . . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) | 
| 3 | numth3 10510 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ dom card) | |
| 4 | 2, 3 | syl 17 | . 2 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ dom card) | 
| 5 | infxpidm2 10057 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴) | |
| 6 | 4, 5 | mpancom 688 | 1 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 × cxp 5683 dom cdm 5685 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 cardccrd 9975 | 
| 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 ax-ac2 10503 | 
| 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 df-ac 10156 | 
| This theorem is referenced by: unirnfdomd 10607 inar1 10815 | 
| Copyright terms: Public domain | W3C validator |