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Mirrors > Home > MPE Home > Th. List > infxpidm2 | Structured version Visualization version GIF version |
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 10559. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
infxpidm2 | β’ ((π΄ β dom card β§ Ο βΌ π΄) β (π΄ Γ π΄) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardid2 9950 | . . . . . 6 β’ (π΄ β dom card β (cardβπ΄) β π΄) | |
2 | 1 | ensymd 9003 | . . . . 5 β’ (π΄ β dom card β π΄ β (cardβπ΄)) |
3 | xpen 9142 | . . . . 5 β’ ((π΄ β (cardβπ΄) β§ π΄ β (cardβπ΄)) β (π΄ Γ π΄) β ((cardβπ΄) Γ (cardβπ΄))) | |
4 | 2, 2, 3 | syl2anc 583 | . . . 4 β’ (π΄ β dom card β (π΄ Γ π΄) β ((cardβπ΄) Γ (cardβπ΄))) |
5 | 4 | adantr 480 | . . 3 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (π΄ Γ π΄) β ((cardβπ΄) Γ (cardβπ΄))) |
6 | cardon 9941 | . . . 4 β’ (cardβπ΄) β On | |
7 | cardom 9983 | . . . . 5 β’ (cardβΟ) = Ο | |
8 | omelon 9643 | . . . . . . . 8 β’ Ο β On | |
9 | onenon 9946 | . . . . . . . 8 β’ (Ο β On β Ο β dom card) | |
10 | 8, 9 | ax-mp 5 | . . . . . . 7 β’ Ο β dom card |
11 | carddom2 9974 | . . . . . . 7 β’ ((Ο β dom card β§ π΄ β dom card) β ((cardβΟ) β (cardβπ΄) β Ο βΌ π΄)) | |
12 | 10, 11 | mpan 687 | . . . . . 6 β’ (π΄ β dom card β ((cardβΟ) β (cardβπ΄) β Ο βΌ π΄)) |
13 | 12 | biimpar 477 | . . . . 5 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (cardβΟ) β (cardβπ΄)) |
14 | 7, 13 | eqsstrrid 4026 | . . . 4 β’ ((π΄ β dom card β§ Ο βΌ π΄) β Ο β (cardβπ΄)) |
15 | infxpen 10011 | . . . 4 β’ (((cardβπ΄) β On β§ Ο β (cardβπ΄)) β ((cardβπ΄) Γ (cardβπ΄)) β (cardβπ΄)) | |
16 | 6, 14, 15 | sylancr 586 | . . 3 β’ ((π΄ β dom card β§ Ο βΌ π΄) β ((cardβπ΄) Γ (cardβπ΄)) β (cardβπ΄)) |
17 | entr 9004 | . . 3 β’ (((π΄ Γ π΄) β ((cardβπ΄) Γ (cardβπ΄)) β§ ((cardβπ΄) Γ (cardβπ΄)) β (cardβπ΄)) β (π΄ Γ π΄) β (cardβπ΄)) | |
18 | 5, 16, 17 | syl2anc 583 | . 2 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (π΄ Γ π΄) β (cardβπ΄)) |
19 | 1 | adantr 480 | . 2 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (cardβπ΄) β π΄) |
20 | entr 9004 | . 2 β’ (((π΄ Γ π΄) β (cardβπ΄) β§ (cardβπ΄) β π΄) β (π΄ Γ π΄) β π΄) | |
21 | 18, 19, 20 | syl2anc 583 | 1 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (π΄ Γ π΄) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β wcel 2098 β wss 3943 class class class wbr 5141 Γ cxp 5667 dom cdm 5669 Oncon0 6358 βcfv 6537 Οcom 7852 β cen 8938 βΌ cdom 8939 cardccrd 9932 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-oi 9507 df-card 9936 |
This theorem is referenced by: infpwfien 10059 mappwen 10109 infdjuabs 10203 infxpdom 10208 fin67 10392 infxpidm 10559 ttac 42353 |
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