Theorem infcda 9030

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
infcda	⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵))

Proof of Theorem infcda

Step	Hyp	Ref	Expression
1		unnum 9022	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card)
2	1	3adant3 1081	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ∈ dom card)
3		ssun2 3777	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
4		ssdomg 8001	. . . . . 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		cdadom2 9009	. . . . 5 ⊢ (𝐵 ≼ (𝐴 ∪ 𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 +𝑐 (𝐴 ∪ 𝐵)))
7	5, 6	syl 17	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≼ (𝐴 +𝑐 (𝐴 ∪ 𝐵)))
8		cdacomen 9003	. . . 4 ⊢ (𝐴 +𝑐 (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) +𝑐 𝐴)
9		domentr 8015	. . . 4 ⊢ (((𝐴 +𝑐 𝐵) ≼ (𝐴 +𝑐 (𝐴 ∪ 𝐵)) ∧ (𝐴 +𝑐 (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) +𝑐 𝐴)) → (𝐴 +𝑐 𝐵) ≼ ((𝐴 ∪ 𝐵) +𝑐 𝐴))
10	7, 8, 9	sylancl 694	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≼ ((𝐴 ∪ 𝐵) +𝑐 𝐴))
11		simp3 1063	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
12		ssun1 3776	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
13		ssdomg 8001	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∈ dom card → (𝐴 ⊆ (𝐴 ∪ 𝐵) → 𝐴 ≼ (𝐴 ∪ 𝐵)))
14	2, 12, 13	mpisyl 21	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (𝐴 ∪ 𝐵))
15		domtr 8009	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → ω ≼ (𝐴 ∪ 𝐵))
16	11, 14, 15	syl2anc 693	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ (𝐴 ∪ 𝐵))
17		infcdaabs 9028	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ dom card ∧ ω ≼ (𝐴 ∪ 𝐵) ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → ((𝐴 ∪ 𝐵) +𝑐 𝐴) ≈ (𝐴 ∪ 𝐵))
18	2, 16, 14, 17	syl3anc 1326	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴 ∪ 𝐵) +𝑐 𝐴) ≈ (𝐴 ∪ 𝐵))
19		domentr 8015	. . 3 ⊢ (((𝐴 +𝑐 𝐵) ≼ ((𝐴 ∪ 𝐵) +𝑐 𝐴) ∧ ((𝐴 ∪ 𝐵) +𝑐 𝐴) ≈ (𝐴 ∪ 𝐵)) → (𝐴 +𝑐 𝐵) ≼ (𝐴 ∪ 𝐵))
20	10, 18, 19	syl2anc 693	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≼ (𝐴 ∪ 𝐵))
21		uncdadom 8993	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵))
22	21	3adant3 1081	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵))
23		sbth 8080	. 2 ⊢ (((𝐴 +𝑐 𝐵) ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵)) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵))
24	20, 22, 23	syl2anc 693	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ w3a 1037   ∈ wcel 1990   ∪ cun 3572   ⊆ wss 3574   class class class wbr 4653  dom cdm 5114  (class class class)co 6650  ωcom 7065   ≈ cen 7952   ≼ cdom 7953  cardccrd 8761   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-cda 8990

This theorem is referenced by:  alephadd  9399