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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 9780 (used via infxpidm2 9783). (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 8726 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex2i 5639 | . . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
3 | numth3 10236 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ dom card) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ dom card) |
5 | infxpidm2 9783 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴) | |
6 | 4, 5 | mpancom 685 | 1 ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3429 class class class wbr 5073 × cxp 5582 dom cdm 5584 ωcom 7702 ≈ cen 8717 ≼ cdom 8718 cardccrd 9703 |
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 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-inf2 9386 ax-ac2 10229 |
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-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 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 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-oi 9256 df-card 9707 df-ac 9882 |
This theorem is referenced by: unirnfdomd 10333 inar1 10541 |
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