Theorem infdju 9623

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 9615	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card)
2	1	3adant3 1127	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ∈ dom card)
3		ssun2 4142	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
4		ssdomg 8548	. . . . . 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 1131	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card)
7		djudom2 9602	. . . . 5 ⊢ ((𝐵 ≼ (𝐴 ∪ 𝐵) ∧ 𝐴 ∈ dom card) → (𝐴 ⊔ 𝐵) ≼ (𝐴 ⊔ (𝐴 ∪ 𝐵)))
8	5, 6, 7	syl2anc 586	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≼ (𝐴 ⊔ (𝐴 ∪ 𝐵)))
9		djucomen 9596	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ (𝐴 ∪ 𝐵) ∈ dom card) → (𝐴 ⊔ (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) ⊔ 𝐴))
10	6, 2, 9	syl2anc 586	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) ⊔ 𝐴))
11		domentr 8561	. . . 4 ⊢ (((𝐴 ⊔ 𝐵) ≼ (𝐴 ⊔ (𝐴 ∪ 𝐵)) ∧ (𝐴 ⊔ (𝐴 ∪ 𝐵)) ≈ ((𝐴 ∪ 𝐵) ⊔ 𝐴)) → (𝐴 ⊔ 𝐵) ≼ ((𝐴 ∪ 𝐵) ⊔ 𝐴))
12	8, 10, 11	syl2anc 586	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≼ ((𝐴 ∪ 𝐵) ⊔ 𝐴))
13		simp3 1133	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
14		ssun1 4141	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
15		ssdomg 8548	. . . . . 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 8555	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → ω ≼ (𝐴 ∪ 𝐵))
18	13, 16, 17	syl2anc 586	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ (𝐴 ∪ 𝐵))
19		infdjuabs 9621	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ dom card ∧ ω ≼ (𝐴 ∪ 𝐵) ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → ((𝐴 ∪ 𝐵) ⊔ 𝐴) ≈ (𝐴 ∪ 𝐵))
20	2, 18, 16, 19	syl3anc 1366	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴 ∪ 𝐵) ⊔ 𝐴) ≈ (𝐴 ∪ 𝐵))
21		domentr 8561	. . 3 ⊢ (((𝐴 ⊔ 𝐵) ≼ ((𝐴 ∪ 𝐵) ⊔ 𝐴) ∧ ((𝐴 ∪ 𝐵) ⊔ 𝐴) ≈ (𝐴 ∪ 𝐵)) → (𝐴 ⊔ 𝐵) ≼ (𝐴 ∪ 𝐵))
22	12, 20, 21	syl2anc 586	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≼ (𝐴 ∪ 𝐵))
23		undjudom 9586	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵))
24	23	3adant3 1127	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵))
25		sbth 8630	. 2 ⊢ (((𝐴 ⊔ 𝐵) ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵))
26	22, 24, 25	syl2anc 586	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ w3a 1082   ∈ wcel 2113   ∪ cun 3927   ⊆ wss 3929   class class class wbr 5059  dom cdm 5548  ωcom 7573   ≈ cen 8499   ≼ cdom 8500   ⊔ cdju 9320  cardccrd 9357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-inf2 9097

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-1st 7682  df-2nd 7683  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-1o 8095  df-2o 8096  df-oadd 8099  df-er 8282  df-en 8503  df-dom 8504  df-sdom 8505  df-fin 8506  df-oi 8967  df-dju 9323  df-card 9361

This theorem is referenced by:  alephadd  9992