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Theorem infxpidm2 10055
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 10600. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infxpidm2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)

Proof of Theorem infxpidm2
StepHypRef Expression
1 cardid2 9991 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
21ensymd 9044 . . . . 5 (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3 xpen 9179 . . . . 5 ((𝐴 ≈ (card‘𝐴) ∧ 𝐴 ≈ (card‘𝐴)) → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
42, 2, 3syl2anc 584 . . . 4 (𝐴 ∈ dom card → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
54adantr 480 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
6 cardon 9982 . . . 4 (card‘𝐴) ∈ On
7 cardom 10024 . . . . 5 (card‘ω) = ω
8 omelon 9684 . . . . . . . 8 ω ∈ On
9 onenon 9987 . . . . . . . 8 (ω ∈ On → ω ∈ dom card)
108, 9ax-mp 5 . . . . . . 7 ω ∈ dom card
11 carddom2 10015 . . . . . . 7 ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
1210, 11mpan 690 . . . . . 6 (𝐴 ∈ dom card → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
1312biimpar 477 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘ω) ⊆ (card‘𝐴))
147, 13eqsstrrid 4045 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ⊆ (card‘𝐴))
15 infxpen 10052 . . . 4 (((card‘𝐴) ∈ On ∧ ω ⊆ (card‘𝐴)) → ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴))
166, 14, 15sylancr 587 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴))
17 entr 9045 . . 3 (((𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)) ∧ ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴)) → (𝐴 × 𝐴) ≈ (card‘𝐴))
185, 16, 17syl2anc 584 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ (card‘𝐴))
191adantr 480 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ≈ 𝐴)
20 entr 9045 . 2 (((𝐴 × 𝐴) ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
2118, 19, 20syl2anc 584 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  wss 3963   class class class wbr 5148   × cxp 5687  dom cdm 5689  Oncon0 6386  cfv 6563  ωcom 7887  cen 8981  cdom 8982  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-card 9977
This theorem is referenced by:  infpwfien  10100  mappwen  10150  infdjuabs  10243  infxpdom  10248  fin67  10433  infxpidm  10600  ttac  43025
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