Theorem infxpidm2 9428

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 9973. (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 9366	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2	1	ensymd 8543	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3		xpen 8664	. . . . 5 ⊢ ((𝐴 ≈ (card‘𝐴) ∧ 𝐴 ≈ (card‘𝐴)) → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
4	2, 2, 3	syl2anc 587	. . . 4 ⊢ (𝐴 ∈ dom card → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
5	4	adantr 484	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)))
6		cardon 9357	. . . 4 ⊢ (card‘𝐴) ∈ On
7		cardom 9399	. . . . 5 ⊢ (card‘ω) = ω
8		omelon 9093	. . . . . . . 8 ⊢ ω ∈ On
9		onenon 9362	. . . . . . . 8 ⊢ (ω ∈ On → ω ∈ dom card)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ ω ∈ dom card
11		carddom2 9390	. . . . . . 7 ⊢ ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
12	10, 11	mpan 689	. . . . . 6 ⊢ (𝐴 ∈ dom card → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
13	12	biimpar 481	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘ω) ⊆ (card‘𝐴))
14	7, 13	eqsstrrid 3964	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ⊆ (card‘𝐴))
15		infxpen 9425	. . . 4 ⊢ (((card‘𝐴) ∈ On ∧ ω ⊆ (card‘𝐴)) → ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴))
16	6, 14, 15	sylancr 590	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴))
17		entr 8544	. . 3 ⊢ (((𝐴 × 𝐴) ≈ ((card‘𝐴) × (card‘𝐴)) ∧ ((card‘𝐴) × (card‘𝐴)) ≈ (card‘𝐴)) → (𝐴 × 𝐴) ≈ (card‘𝐴))
18	5, 16, 17	syl2anc 587	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ (card‘𝐴))
19	1	adantr 484	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ≈ 𝐴)
20		entr 8544	. 2 ⊢ (((𝐴 × 𝐴) ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
21	18, 19, 20	syl2anc 587	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111   ⊆ wss 3881   class class class wbr 5030   × cxp 5517  dom cdm 5519  Oncon0 6159  ‘cfv 6324  ωcom 7560   ≈ cen 8489   ≼ cdom 8490  cardccrd 9348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-oi 8958  df-card 9352

This theorem is referenced by:  infpwfien  9473  mappwen  9523  infdjuabs  9617  infxpdom  9622  fin67  9806  infxpidm  9973  ttac  39977