MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infxpidm2 Structured version   Visualization version   GIF version

Theorem infxpidm2 9960
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 10505. (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 9896 . . . . . 6 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
21ensymd 8952 . . . . 5 (𝐴 ∈ dom card β†’ 𝐴 β‰ˆ (cardβ€˜π΄))
3 xpen 9091 . . . . 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 585 . . . 4 (𝐴 ∈ dom card β†’ (𝐴 Γ— 𝐴) β‰ˆ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)))
54adantr 482 . . 3 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝐴 Γ— 𝐴) β‰ˆ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)))
6 cardon 9887 . . . 4 (cardβ€˜π΄) ∈ On
7 cardom 9929 . . . . 5 (cardβ€˜Ο‰) = Ο‰
8 omelon 9589 . . . . . . . 8 Ο‰ ∈ On
9 onenon 9892 . . . . . . . 8 (Ο‰ ∈ On β†’ Ο‰ ∈ dom card)
108, 9ax-mp 5 . . . . . . 7 Ο‰ ∈ dom card
11 carddom2 9920 . . . . . . 7 ((Ο‰ ∈ dom card ∧ 𝐴 ∈ dom card) β†’ ((cardβ€˜Ο‰) βŠ† (cardβ€˜π΄) ↔ Ο‰ β‰Ό 𝐴))
1210, 11mpan 689 . . . . . 6 (𝐴 ∈ dom card β†’ ((cardβ€˜Ο‰) βŠ† (cardβ€˜π΄) ↔ Ο‰ β‰Ό 𝐴))
1312biimpar 479 . . . . 5 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (cardβ€˜Ο‰) βŠ† (cardβ€˜π΄))
147, 13eqsstrrid 3998 . . . 4 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ Ο‰ βŠ† (cardβ€˜π΄))
15 infxpen 9957 . . . 4 (((cardβ€˜π΄) ∈ On ∧ Ο‰ βŠ† (cardβ€˜π΄)) β†’ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)) β‰ˆ (cardβ€˜π΄))
166, 14, 15sylancr 588 . . 3 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ ((cardβ€˜π΄) Γ— (cardβ€˜π΄)) β‰ˆ (cardβ€˜π΄))
17 entr 8953 . . 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 585 . 2 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝐴 Γ— 𝐴) β‰ˆ (cardβ€˜π΄))
191adantr 482 . 2 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
20 entr 8953 . 2 (((𝐴 Γ— 𝐴) β‰ˆ (cardβ€˜π΄) ∧ (cardβ€˜π΄) β‰ˆ 𝐴) β†’ (𝐴 Γ— 𝐴) β‰ˆ 𝐴)
2118, 19, 20syl2anc 585 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 397   ∈ wcel 2107   βŠ† wss 3915   class class class wbr 5110   Γ— cxp 5636  dom cdm 5638  Oncon0 6322  β€˜cfv 6501  Ο‰com 7807   β‰ˆ cen 8887   β‰Ό cdom 8888  cardccrd 9878
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9453  df-card 9882
This theorem is referenced by:  infpwfien  10005  mappwen  10055  infdjuabs  10149  infxpdom  10154  fin67  10338  infxpidm  10505  ttac  41389
  Copyright terms: Public domain W3C validator