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Theorem infxpidm2 9127
Description: The Cartesian product of an infinite set with itself is idempotent. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 9673. (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 9066 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
21ensymd 8247 . . . . 5 (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3 xpen 8366 . . . . 5 ((𝐴 ≈ (card‘𝐴) ∧ 𝐴 ≈ (card‘𝐴)) → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
42, 2, 3syl2anc 580 . . . 4 (𝐴 ∈ dom card → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
54adantr 473 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
6 cardon 9057 . . . 4 (card‘𝐴) ∈ On
7 cardom 9099 . . . . 5 (card‘ω) = ω
8 omelon 8794 . . . . . . . 8 ω ∈ On
9 onenon 9062 . . . . . . . 8 (ω ∈ On → ω ∈ dom card)
108, 9ax-mp 5 . . . . . . 7 ω ∈ dom card
11 carddom2 9090 . . . . . . 7 ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
1210, 11mpan 682 . . . . . 6 (𝐴 ∈ dom card → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
1312biimpar 470 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘ω) ⊆ (card‘𝐴))
147, 13syl5eqssr 3847 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ⊆ (card‘𝐴))
15 infxpen 9124 . . . 4 (((card‘𝐴) ∈ On ∧ ω ⊆ (card‘𝐴)) → ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴))
166, 14, 15sylancr 582 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴))
17 entr 8248 . . 3 (((𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)) ∧ ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴)) → (𝐴 × 𝐴) ≈ (card‘𝐴))
185, 16, 17syl2anc 580 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ (card‘𝐴))
191adantr 473 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ≈ 𝐴)
20 entr 8248 . 2 (((𝐴 × 𝐴) ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
2118, 19, 20syl2anc 580 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wcel 2157  wss 3770   class class class wbr 4844   × cxp 5311  dom cdm 5313  Oncon0 5942  cfv 6102  ωcom 7300  cen 8193  cdom 8194  cardccrd 9048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184  ax-inf2 8789
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-se 5273  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-pred 5899  df-ord 5945  df-on 5946  df-lim 5947  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-isom 6111  df-riota 6840  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-om 7301  df-1st 7402  df-2nd 7403  df-wrecs 7646  df-recs 7708  df-rdg 7746  df-1o 7800  df-oadd 7804  df-er 7983  df-en 8197  df-dom 8198  df-sdom 8199  df-fin 8200  df-oi 8658  df-card 9052
This theorem is referenced by:  infpwfien  9172  mappwen  9222  infcdaabs  9317  infxpdom  9322  fin67  9506  infxpidm  9673  ttac  38383
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