Theorem infdju 10290

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 10280	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card)
2	1	3adant3 1150	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ∈ dom card)
3		ssun2 4176	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
4		ssdomg 9032	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∈ dom card → (𝐵 ⊆ (𝐴 ∪ 𝐵) → 𝐵 ≼ (𝐴 ∪ 𝐵)))
5	2, 3, 4	mpisyl 22	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐵 ≼ (𝐴 ∪ 𝐵))
6		simp1 1154	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card)
7		djudom2 10267	. . . . 5 ⊢ ((𝐵 ≼ (𝐴 ∪ 𝐵) ∧ 𝐴 ∈ dom card) → (𝐴 ⊔ 𝐵) ≼ (𝐴 ⊔ (𝐴 ∪ 𝐵)))
8	5, 6, 7	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≼ (𝐴 ⊔ (𝐴 ∪ 𝐵)))
9		djucomen 10261	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ (𝐴 ∪ 𝐵) ∈ dom card) → (𝐴 ⊔ (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) ⊔ 𝐴))
10	6, 2, 9	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) ⊔ 𝐴))
11		domentr 9045	. . . 4 ⊢ (((𝐴 ⊔ 𝐵) ≼ (𝐴 ⊔ (𝐴 ∪ 𝐵)) ∧ (𝐴 ⊔ (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) ⊔ 𝐴)) → (𝐴 ⊔ 𝐵) ≼ ((𝐴 ∪ 𝐵) ⊔ 𝐴))
12	8, 10, 11	syl2anc 596	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≼ ((𝐴 ∪ 𝐵) ⊔ 𝐴))
13		simp3 1156	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
14		ssun1 4175	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
15		ssdomg 9032	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∈ dom card → (𝐴 ⊆ (𝐴 ∪ 𝐵) → 𝐴 ≼ (𝐴 ∪ 𝐵)))
16	2, 14, 15	mpisyl 22	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (𝐴 ∪ 𝐵))
17		domtr 9039	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → ω ≼ (𝐴 ∪ 𝐵))
18	13, 16, 17	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ (𝐴 ∪ 𝐵))
19		infdjuabs 10288	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ dom card ∧ ω ≼ (𝐴 ∪ 𝐵) ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → ((𝐴 ∪ 𝐵) ⊔ 𝐴) ≈ (𝐴 ∪ 𝐵))
20	2, 18, 16, 19	syl3anc 1398	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴 ∪ 𝐵) ⊔ 𝐴) ≈ (𝐴 ∪ 𝐵))
21		domentr 9045	. . 3 ⊢ (((𝐴 ⊔ 𝐵) ≼ ((𝐴 ∪ 𝐵) ⊔ 𝐴) ∧ ((𝐴 ∪ 𝐵) ⊔ 𝐴) ≈ (𝐴 ∪ 𝐵)) → (𝐴 ⊔ 𝐵) ≼ (𝐴 ∪ 𝐵))
22	12, 20, 21	syl2anc 596	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≼ (𝐴 ∪ 𝐵))
23		undjudom 10251	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵))
24	23	3adant3 1150	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵))
25		sbth 9121	. 2 ⊢ (((𝐴 ⊔ 𝐵) ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵))
26	22, 24, 25	syl2anc 596	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ wcel 2144   ⊆ wss 3896   ∪ cun 4148   class class class wbr 5102  dom cdm 5650  ωcom 7879   ≈ cen 8975   ≼ cdom 8976   ⊔ cdju 9984  cardccrd 10021

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 7753  ax-inf2 9647

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 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  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 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-2o 8482  df-oadd 8485  df-er 8722  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-oi 9509  df-dju 9987  df-card 10025

This theorem is used by:  alephadd  10667