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Theorem infxpidm2 10040
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 10585. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infxpidm2 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝐴 Γ— 𝐴) β‰ˆ 𝐴)

Proof of Theorem infxpidm2
StepHypRef Expression
1 cardid2 9976 . . . . . 6 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
21ensymd 9024 . . . . 5 (𝐴 ∈ dom card β†’ 𝐴 β‰ˆ (cardβ€˜π΄))
3 xpen 9163 . . . . 5 ((𝐴 β‰ˆ (cardβ€˜π΄) ∧ 𝐴 β‰ˆ (cardβ€˜π΄)) β†’ (𝐴 Γ— 𝐴) β‰ˆ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)))
42, 2, 3syl2anc 582 . . . 4 (𝐴 ∈ dom card β†’ (𝐴 Γ— 𝐴) β‰ˆ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)))
54adantr 479 . . 3 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝐴 Γ— 𝐴) β‰ˆ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)))
6 cardon 9967 . . . 4 (cardβ€˜π΄) ∈ On
7 cardom 10009 . . . . 5 (cardβ€˜Ο‰) = Ο‰
8 omelon 9669 . . . . . . . 8 Ο‰ ∈ On
9 onenon 9972 . . . . . . . 8 (Ο‰ ∈ On β†’ Ο‰ ∈ dom card)
108, 9ax-mp 5 . . . . . . 7 Ο‰ ∈ dom card
11 carddom2 10000 . . . . . . 7 ((Ο‰ ∈ dom card ∧ 𝐴 ∈ dom card) β†’ ((cardβ€˜Ο‰) βŠ† (cardβ€˜π΄) ↔ Ο‰ β‰Ό 𝐴))
1210, 11mpan 688 . . . . . 6 (𝐴 ∈ dom card β†’ ((cardβ€˜Ο‰) βŠ† (cardβ€˜π΄) ↔ Ο‰ β‰Ό 𝐴))
1312biimpar 476 . . . . 5 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (cardβ€˜Ο‰) βŠ† (cardβ€˜π΄))
147, 13eqsstrrid 4022 . . . 4 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ Ο‰ βŠ† (cardβ€˜π΄))
15 infxpen 10037 . . . 4 (((cardβ€˜π΄) ∈ On ∧ Ο‰ βŠ† (cardβ€˜π΄)) β†’ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)) β‰ˆ (cardβ€˜π΄))
166, 14, 15sylancr 585 . . 3 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)) β‰ˆ (cardβ€˜π΄))
17 entr 9025 . . 3 (((𝐴 Γ— 𝐴) β‰ˆ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)) ∧ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)) β‰ˆ (cardβ€˜π΄)) β†’ (𝐴 Γ— 𝐴) β‰ˆ (cardβ€˜π΄))
185, 16, 17syl2anc 582 . 2 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝐴 Γ— 𝐴) β‰ˆ (cardβ€˜π΄))
191adantr 479 . 2 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
20 entr 9025 . 2 (((𝐴 Γ— 𝐴) β‰ˆ (cardβ€˜π΄) ∧ (cardβ€˜π΄) β‰ˆ 𝐴) β†’ (𝐴 Γ— 𝐴) β‰ˆ 𝐴)
2118, 19, 20syl2anc 582 1 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝐴 Γ— 𝐴) β‰ˆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∈ wcel 2098   βŠ† wss 3939   class class class wbr 5143   Γ— cxp 5670  dom cdm 5672  Oncon0 6364  β€˜cfv 6543  Ο‰com 7868   β‰ˆ cen 8959   β‰Ό cdom 8960  cardccrd 9958
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7372  df-ov 7419  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-oi 9533  df-card 9962
This theorem is referenced by:  infpwfien  10085  mappwen  10135  infdjuabs  10229  infxpdom  10234  fin67  10418  infxpidm  10585  ttac  42522
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