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Theorem infdju 9622
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) → (𝐴𝐵) ≈ (𝐴𝐵))

Proof of Theorem infdju
StepHypRef Expression
1 unnum 9614 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ∈ dom card)
213adant3 1127 . . . . . 6 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵) ∈ dom card)
3 ssun2 4147 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
4 ssdomg 8547 . . . . . 6 ((𝐴𝐵) ∈ dom card → (𝐵 ⊆ (𝐴𝐵) → 𝐵 ≼ (𝐴𝐵)))
52, 3, 4mpisyl 21 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐵 ≼ (𝐴𝐵))
6 simp1 1131 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card)
7 djudom2 9601 . . . . 5 ((𝐵 ≼ (𝐴𝐵) ∧ 𝐴 ∈ dom card) → (𝐴𝐵) ≼ (𝐴 ⊔ (𝐴𝐵)))
85, 6, 7syl2anc 586 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵) ≼ (𝐴 ⊔ (𝐴𝐵)))
9 djucomen 9595 . . . . 5 ((𝐴 ∈ dom card ∧ (𝐴𝐵) ∈ dom card) → (𝐴 ⊔ (𝐴𝐵)) ≈ ((𝐴𝐵) ⊔ 𝐴))
106, 2, 9syl2anc 586 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ (𝐴𝐵)) ≈ ((𝐴𝐵) ⊔ 𝐴))
11 domentr 8560 . . . 4 (((𝐴𝐵) ≼ (𝐴 ⊔ (𝐴𝐵)) ∧ (𝐴 ⊔ (𝐴𝐵)) ≈ ((𝐴𝐵) ⊔ 𝐴)) → (𝐴𝐵) ≼ ((𝐴𝐵) ⊔ 𝐴))
128, 10, 11syl2anc 586 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵) ≼ ((𝐴𝐵) ⊔ 𝐴))
13 simp3 1133 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
14 ssun1 4146 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
15 ssdomg 8547 . . . . . 6 ((𝐴𝐵) ∈ dom card → (𝐴 ⊆ (𝐴𝐵) → 𝐴 ≼ (𝐴𝐵)))
162, 14, 15mpisyl 21 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (𝐴𝐵))
17 domtr 8554 . . . . 5 ((ω ≼ 𝐴𝐴 ≼ (𝐴𝐵)) → ω ≼ (𝐴𝐵))
1813, 16, 17syl2anc 586 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ (𝐴𝐵))
19 infdjuabs 9620 . . . 4 (((𝐴𝐵) ∈ dom card ∧ ω ≼ (𝐴𝐵) ∧ 𝐴 ≼ (𝐴𝐵)) → ((𝐴𝐵) ⊔ 𝐴) ≈ (𝐴𝐵))
202, 18, 16, 19syl3anc 1366 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) ⊔ 𝐴) ≈ (𝐴𝐵))
21 domentr 8560 . . 3 (((𝐴𝐵) ≼ ((𝐴𝐵) ⊔ 𝐴) ∧ ((𝐴𝐵) ⊔ 𝐴) ≈ (𝐴𝐵)) → (𝐴𝐵) ≼ (𝐴𝐵))
2212, 20, 21syl2anc 586 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵) ≼ (𝐴𝐵))
23 undjudom 9585 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ≼ (𝐴𝐵))
24233adant3 1127 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵) ≼ (𝐴𝐵))
25 sbth 8629 . 2 (((𝐴𝐵) ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ (𝐴𝐵)) → (𝐴𝐵) ≈ (𝐴𝐵))
2622, 24, 25syl2anc 586 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵) ≈ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1082  wcel 2108  cun 3932  wss 3934   class class class wbr 5057  dom cdm 5548  ωcom 7572  cen 8498  cdom 8499  cdju 9319  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  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 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-oi 8966  df-dju 9322  df-card 9360
This theorem is referenced by:  alephadd  9991
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