Theorem infcda 9068

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 9060	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card)
2	1	3adant3 1101	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ∈ dom card)
3		ssun2 3810	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
4		ssdomg 8043	. . . . . 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 9047	. . . . 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 9041	. . . 4 ⊢ (𝐴 +𝑐 (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) +𝑐 𝐴)
9		domentr 8056	. . . 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 695	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≼ ((𝐴 ∪ 𝐵) +𝑐 𝐴))
11		simp3 1083	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
12		ssun1 3809	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
13		ssdomg 8043	. . . . . 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 8050	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → ω ≼ (𝐴 ∪ 𝐵))
16	11, 14, 15	syl2anc 694	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ (𝐴 ∪ 𝐵))
17		infcdaabs 9066	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ dom card ∧ ω ≼ (𝐴 ∪ 𝐵) ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → ((𝐴 ∪ 𝐵) +𝑐 𝐴) ≈ (𝐴 ∪ 𝐵))
18	2, 16, 14, 17	syl3anc 1366	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴 ∪ 𝐵) +𝑐 𝐴) ≈ (𝐴 ∪ 𝐵))
19		domentr 8056	. . 3 ⊢ (((𝐴 +𝑐 𝐵) ≼ ((𝐴 ∪ 𝐵) +𝑐 𝐴) ∧ ((𝐴 ∪ 𝐵) +𝑐 𝐴) ≈ (𝐴 ∪ 𝐵)) → (𝐴 +𝑐 𝐵) ≼ (𝐴 ∪ 𝐵))
20	10, 18, 19	syl2anc 694	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≼ (𝐴 ∪ 𝐵))
21		uncdadom 9031	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵))
22	21	3adant3 1101	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵))
23		sbth 8121	. 2 ⊢ (((𝐴 +𝑐 𝐵) ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵)) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵))
24	20, 22, 23	syl2anc 694	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ w3a 1054   ∈ wcel 2030   ∪ cun 3605   ⊆ wss 3607   class class class wbr 4685  dom cdm 5143  (class class class)co 6690  ωcom 7107   ≈ cen 7994   ≼ cdom 7995  cardccrd 8799   +_𝑐 ccda 9027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-card 8803  df-cda 9028

This theorem is referenced by:  alephadd  9437