Theorem infdju 10219

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 10209	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card)
2	1	3adant3 1132	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ∈ dom card)
3		ssun2 4154	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
4		ssdomg 9012	. . . . . 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 1136	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card)
7		djudom2 10196	. . . . 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 10190	. . . . 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 9025	. . . 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 1138	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
14		ssun1 4153	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
15		ssdomg 9012	. . . . . 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 9019	. . . . 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 10217	. . . 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 9025	. . 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 10180	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵))
24	23	3adant3 1132	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵))
25		sbth 9105	. 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 1086   ∈ wcel 2108   ∪ cun 3924   ⊆ wss 3926   class class class wbr 5119  dom cdm 5654  ωcom 7859   ≈ cen 8954   ≼ cdom 8955   ⊔ cdju 9910  cardccrd 9947

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-inf2 9653

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-isom 6539  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-oadd 8482  df-er 8717  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-oi 9522  df-dju 9913  df-card 9951

This theorem is referenced by:  alephadd  10589