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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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