Theorem infdju 10109

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 10099	. . . . . . 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 4128	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
4		ssdomg 8933	. . . . . 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 10086	. . . . 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 10080	. . . . 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 8946	. . . 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 4127	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
15		ssdomg 8933	. . . . . 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 8940	. . . . 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 10107	. . . 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 8946	. . 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 10070	. . 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 9021	. 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 2113   ∪ cun 3896   ⊆ wss 3898   class class class wbr 5095  dom cdm 5621  ωcom 7805   ≈ cen 8876   ≼ cdom 8877   ⊔ cdju 9802  cardccrd 9839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-inf2 9542

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-oadd 8398  df-er 8631  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-oi 9407  df-dju 9805  df-card 9843

This theorem is referenced by:  alephadd  10479