Theorem infdju 10247

Description: The sum of two cardinal numbers is their maximum, if one of them is infinite. Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
infdju	⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵))

Proof of Theorem infdju

Step	Hyp	Ref	Expression
1		unnum 10237	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card)
2	1	3adant3 1133	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ∈ dom card)
3		ssun2 4179	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
4		ssdomg 9040	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∈ dom card → (𝐵 ⊆ (𝐴 ∪ 𝐵) → 𝐵 ≼ (𝐴 ∪ 𝐵)))
5	2, 3, 4	mpisyl 21	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐵 ≼ (𝐴 ∪ 𝐵))
6		simp1 1137	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card)
7		djudom2 10224	. . . . 5 ⊢ ((𝐵 ≼ (𝐴 ∪ 𝐵) ∧ 𝐴 ∈ dom card) → (𝐴 ⊔ 𝐵) ≼ (𝐴 ⊔ (𝐴 ∪ 𝐵)))
8	5, 6, 7	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≼ (𝐴 ⊔ (𝐴 ∪ 𝐵)))
9		djucomen 10218	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ (𝐴 ∪ 𝐵) ∈ dom card) → (𝐴 ⊔ (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) ⊔ 𝐴))
10	6, 2, 9	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) ⊔ 𝐴))
11		domentr 9053	. . . 4 ⊢ (((𝐴 ⊔ 𝐵) ≼ (𝐴 ⊔ (𝐴 ∪ 𝐵)) ∧ (𝐴 ⊔ (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) ⊔ 𝐴)) → (𝐴 ⊔ 𝐵) ≼ ((𝐴 ∪ 𝐵) ⊔ 𝐴))
12	8, 10, 11	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≼ ((𝐴 ∪ 𝐵) ⊔ 𝐴))
13		simp3 1139	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
14		ssun1 4178	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
15		ssdomg 9040	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∈ dom card → (𝐴 ⊆ (𝐴 ∪ 𝐵) → 𝐴 ≼ (𝐴 ∪ 𝐵)))
16	2, 14, 15	mpisyl 21	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (𝐴 ∪ 𝐵))
17		domtr 9047	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → ω ≼ (𝐴 ∪ 𝐵))
18	13, 16, 17	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ (𝐴 ∪ 𝐵))
19		infdjuabs 10245	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ dom card ∧ ω ≼ (𝐴 ∪ 𝐵) ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → ((𝐴 ∪ 𝐵) ⊔ 𝐴) ≈ (𝐴 ∪ 𝐵))
20	2, 18, 16, 19	syl3anc 1373	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴 ∪ 𝐵) ⊔ 𝐴) ≈ (𝐴 ∪ 𝐵))
21		domentr 9053	. . 3 ⊢ (((𝐴 ⊔ 𝐵) ≼ ((𝐴 ∪ 𝐵) ⊔ 𝐴) ∧ ((𝐴 ∪ 𝐵) ⊔ 𝐴) ≈ (𝐴 ∪ 𝐵)) → (𝐴 ⊔ 𝐵) ≼ (𝐴 ∪ 𝐵))
22	12, 20, 21	syl2anc 584	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≼ (𝐴 ∪ 𝐵))
23		undjudom 10208	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵))
24	23	3adant3 1133	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵))
25		sbth 9133	. 2 ⊢ (((𝐴 ⊔ 𝐵) ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵))
26	22, 24, 25	syl2anc 584	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2108   ∪ cun 3949   ⊆ wss 3951   class class class wbr 5143  dom cdm 5685  ωcom 7887   ≈ cen 8982   ≼ cdom 8983   ⊔ cdju 9938  cardccrd 9975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-dju 9941  df-card 9979

This theorem is referenced by:  alephadd  10617