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Theorem infxpidm2 10014
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.)
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 9950 . . . . . 6 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
21ensymd 9003 . . . . 5 (𝐴 ∈ dom card β†’ 𝐴 β‰ˆ (cardβ€˜π΄))
3 xpen 9142 . . . . 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 583 . . . 4 (𝐴 ∈ dom card β†’ (𝐴 Γ— 𝐴) β‰ˆ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)))
54adantr 480 . . 3 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝐴 Γ— 𝐴) β‰ˆ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)))
6 cardon 9941 . . . 4 (cardβ€˜π΄) ∈ On
7 cardom 9983 . . . . 5 (cardβ€˜Ο‰) = Ο‰
8 omelon 9643 . . . . . . . 8 Ο‰ ∈ On
9 onenon 9946 . . . . . . . 8 (Ο‰ ∈ On β†’ Ο‰ ∈ dom card)
108, 9ax-mp 5 . . . . . . 7 Ο‰ ∈ dom card
11 carddom2 9974 . . . . . . 7 ((Ο‰ ∈ dom card ∧ 𝐴 ∈ dom card) β†’ ((cardβ€˜Ο‰) βŠ† (cardβ€˜π΄) ↔ Ο‰ β‰Ό 𝐴))
1210, 11mpan 687 . . . . . 6 (𝐴 ∈ dom card β†’ ((cardβ€˜Ο‰) βŠ† (cardβ€˜π΄) ↔ Ο‰ β‰Ό 𝐴))
1312biimpar 477 . . . . 5 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (cardβ€˜Ο‰) βŠ† (cardβ€˜π΄))
147, 13eqsstrrid 4026 . . . 4 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ Ο‰ βŠ† (cardβ€˜π΄))
15 infxpen 10011 . . . 4 (((cardβ€˜π΄) ∈ On ∧ Ο‰ βŠ† (cardβ€˜π΄)) β†’ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)) β‰ˆ (cardβ€˜π΄))
166, 14, 15sylancr 586 . . 3 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)) β‰ˆ (cardβ€˜π΄))
17 entr 9004 . . 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 583 . 2 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝐴 Γ— 𝐴) β‰ˆ (cardβ€˜π΄))
191adantr 480 . 2 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
20 entr 9004 . 2 (((𝐴 Γ— 𝐴) β‰ˆ (cardβ€˜π΄) ∧ (cardβ€˜π΄) β‰ˆ 𝐴) β†’ (𝐴 Γ— 𝐴) β‰ˆ 𝐴)
2118, 19, 20syl2anc 583 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 395   ∈ wcel 2098   βŠ† wss 3943   class class class wbr 5141   Γ— cxp 5667  dom cdm 5669  Oncon0 6358  β€˜cfv 6537  Ο‰com 7852   β‰ˆ cen 8938   β‰Ό 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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