Theorem infxpidm2 10100

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 10650. (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

Step	Hyp	Ref	Expression
1		cardid2 10038	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2	1	ensymd 9036	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3		xpen 9163	. . . . 5 ⊢ ((𝐴 ≈ (card‘𝐴) ∧ 𝐴 ≈ (card‘𝐴)) → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
4	2, 2, 3	syl2anc 596	. . . 4 ⊢ (𝐴 ∈ dom card → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
5	4	adantr 486	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
6		cardon 10029	. . . 4 ⊢ (card‘𝐴) ∈ On
7		cardom 10071	. . . . 5 ⊢ (card‘ω) = ω
8		omelon 9651	. . . . . . . 8 ⊢ ω ∈ On
9		onenon 10034	. . . . . . . 8 ⊢ (ω ∈ On → ω ∈ dom card)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ ω ∈ dom card
11		carddom2 10062	. . . . . . 7 ⊢ ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
12	10, 11	mpan 703	. . . . . 6 ⊢ (𝐴 ∈ dom card → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
13	12	biimpar 483	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘ω) ⊆ (card‘𝐴))
14	7, 13	eqsstrrid 3947	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ⊆ (card‘𝐴))
15		infxpen 10097	. . . 4 ⊢ (((card‘𝐴) ∈ On ∧ ω ⊆ (card‘𝐴)) → ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴))
16	6, 14, 15	sylancr 599	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴))
17		entr 9037	. . 3 ⊢ (((𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)) ∧ ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴)) → (𝐴 × 𝐴) ≈ (card‘𝐴))
18	5, 16, 17	syl2anc 596	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ (card‘𝐴))
19	1	adantr 486	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ≈ 𝐴)
20		entr 9037	. 2 ⊢ (((𝐴 × 𝐴) ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
21	18, 19, 20	syl2anc 596	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144   ⊆ wss 3895   class class class wbr 5102   × cxp 5648  dom cdm 5650  Oncon0 6362  ‘cfv 6539  ωcom 7878   ≈ cen 8974   ≼ cdom 8975  cardccrd 10020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-inf2 9646

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7378  df-ov 7424  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-er 8721  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-oi 9508  df-card 10024

This theorem is used by:  infpwfien  10145  mappwen  10195  infdjuabs  10287  infxpdom  10292  fin67  10477  infxpidm  10650  weexenwe  35813  j0ltaleph  35948  ttac  44137