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Theorem infcda 9068
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) ≈ (𝐴𝐵))

Proof of Theorem infcda
StepHypRef Expression
1 unnum 9060 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ∈ dom card)
213adant3 1101 . . . . . 6 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵) ∈ dom card)
3 ssun2 3810 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
4 ssdomg 8043 . . . . . 6 ((𝐴𝐵) ∈ dom card → (𝐵 ⊆ (𝐴𝐵) → 𝐵 ≼ (𝐴𝐵)))
52, 3, 4mpisyl 21 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐵 ≼ (𝐴𝐵))
6 cdadom2 9047 . . . . 5 (𝐵 ≼ (𝐴𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 +𝑐 (𝐴𝐵)))
75, 6syl 17 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≼ (𝐴 +𝑐 (𝐴𝐵)))
8 cdacomen 9041 . . . 4 (𝐴 +𝑐 (𝐴𝐵)) ≈ ((𝐴𝐵) +𝑐 𝐴)
9 domentr 8056 . . . 4 (((𝐴 +𝑐 𝐵) ≼ (𝐴 +𝑐 (𝐴𝐵)) ∧ (𝐴 +𝑐 (𝐴𝐵)) ≈ ((𝐴𝐵) +𝑐 𝐴)) → (𝐴 +𝑐 𝐵) ≼ ((𝐴𝐵) +𝑐 𝐴))
107, 8, 9sylancl 695 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≼ ((𝐴𝐵) +𝑐 𝐴))
11 simp3 1083 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
12 ssun1 3809 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
13 ssdomg 8043 . . . . . 6 ((𝐴𝐵) ∈ dom card → (𝐴 ⊆ (𝐴𝐵) → 𝐴 ≼ (𝐴𝐵)))
142, 12, 13mpisyl 21 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (𝐴𝐵))
15 domtr 8050 . . . . 5 ((ω ≼ 𝐴𝐴 ≼ (𝐴𝐵)) → ω ≼ (𝐴𝐵))
1611, 14, 15syl2anc 694 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ (𝐴𝐵))
17 infcdaabs 9066 . . . 4 (((𝐴𝐵) ∈ dom card ∧ ω ≼ (𝐴𝐵) ∧ 𝐴 ≼ (𝐴𝐵)) → ((𝐴𝐵) +𝑐 𝐴) ≈ (𝐴𝐵))
182, 16, 14, 17syl3anc 1366 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) +𝑐 𝐴) ≈ (𝐴𝐵))
19 domentr 8056 . . 3 (((𝐴 +𝑐 𝐵) ≼ ((𝐴𝐵) +𝑐 𝐴) ∧ ((𝐴𝐵) +𝑐 𝐴) ≈ (𝐴𝐵)) → (𝐴 +𝑐 𝐵) ≼ (𝐴𝐵))
2010, 18, 19syl2anc 694 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≼ (𝐴𝐵))
21 uncdadom 9031 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ≼ (𝐴 +𝑐 𝐵))
22213adant3 1101 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵) ≼ (𝐴 +𝑐 𝐵))
23 sbth 8121 . 2 (((𝐴 +𝑐 𝐵) ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ (𝐴 +𝑐 𝐵)) → (𝐴 +𝑐 𝐵) ≈ (𝐴𝐵))
2420, 22, 23syl2anc 694 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≈ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054  wcel 2030  cun 3605  wss 3607   class class class wbr 4685  dom cdm 5143  (class class class)co 6690  ωcom 7107  cen 7994  cdom 7995  cardccrd 8799   +𝑐 ccda 9027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-card 8803  df-cda 9028
This theorem is referenced by:  alephadd  9437
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