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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 10045 (used via infxpidm2 10048). (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 8976 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex2i 5739 | . . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
3 | numth3 10501 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ dom card) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ dom card) |
5 | infxpidm2 10048 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴) | |
6 | 4, 5 | mpancom 686 | 1 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3473 class class class wbr 5152 × cxp 5680 dom cdm 5682 ωcom 7876 ≈ cen 8967 ≼ cdom 8968 cardccrd 9966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-ac2 10494 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-oi 9541 df-card 9970 df-ac 10147 |
This theorem is referenced by: unirnfdomd 10598 inar1 10806 |
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