Theorem infcda 8913

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 8905	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card)
2	1	3adant3 1074	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ∈ dom card)
3		ssun2 3739	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
4		ssdomg 7887	. . . . . 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 8892	. . . . 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 8886	. . . 4 ⊢ (𝐴 +𝑐 (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) +𝑐 𝐴)
9		domentr 7901	. . . 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 693	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≼ ((𝐴 ∪ 𝐵) +𝑐 𝐴))
11		simp3 1056	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
12		ssun1 3738	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
13		ssdomg 7887	. . . . . 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 7895	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → ω ≼ (𝐴 ∪ 𝐵))
16	11, 14, 15	syl2anc 691	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ (𝐴 ∪ 𝐵))
17		infcdaabs 8911	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ dom card ∧ ω ≼ (𝐴 ∪ 𝐵) ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → ((𝐴 ∪ 𝐵) +𝑐 𝐴) ≈ (𝐴 ∪ 𝐵))
18	2, 16, 14, 17	syl3anc 1318	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴 ∪ 𝐵) +𝑐 𝐴) ≈ (𝐴 ∪ 𝐵))
19		domentr 7901	. . 3 ⊢ (((𝐴 +𝑐 𝐵) ≼ ((𝐴 ∪ 𝐵) +𝑐 𝐴) ∧ ((𝐴 ∪ 𝐵) +𝑐 𝐴) ≈ (𝐴 ∪ 𝐵)) → (𝐴 +𝑐 𝐵) ≼ (𝐴 ∪ 𝐵))
20	10, 18, 19	syl2anc 691	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≼ (𝐴 ∪ 𝐵))
21		uncdadom 8876	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵))
22	21	3adant3 1074	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵))
23		sbth 7965	. 2 ⊢ (((𝐴 +𝑐 𝐵) ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵)) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵))
24	20, 22, 23	syl2anc 691	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ w3a 1031   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038  (class class class)co 6549  ωcom 6957   ≈ cen 7838   ≼ cdom 7839  cardccrd 8644   +_𝑐 ccda 8872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-cda 8873

This theorem is referenced by:  alephadd  9278