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Theorem infcda 9231
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 9223 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ∈ dom card)
213adant3 1125 . . . . . 6 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵) ∈ dom card)
3 ssun2 3926 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
4 ssdomg 8154 . . . . . 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 9210 . . . . 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 9204 . . . 4 (𝐴 +𝑐 (𝐴𝐵)) ≈ ((𝐴𝐵) +𝑐 𝐴)
9 domentr 8167 . . . 4 (((𝐴 +𝑐 𝐵) ≼ (𝐴 +𝑐 (𝐴𝐵)) ∧ (𝐴 +𝑐 (𝐴𝐵)) ≈ ((𝐴𝐵) +𝑐 𝐴)) → (𝐴 +𝑐 𝐵) ≼ ((𝐴𝐵) +𝑐 𝐴))
107, 8, 9sylancl 566 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≼ ((𝐴𝐵) +𝑐 𝐴))
11 simp3 1131 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
12 ssun1 3925 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
13 ssdomg 8154 . . . . . 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 8161 . . . . 5 ((ω ≼ 𝐴𝐴 ≼ (𝐴𝐵)) → ω ≼ (𝐴𝐵))
1611, 14, 15syl2anc 565 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ (𝐴𝐵))
17 infcdaabs 9229 . . . 4 (((𝐴𝐵) ∈ dom card ∧ ω ≼ (𝐴𝐵) ∧ 𝐴 ≼ (𝐴𝐵)) → ((𝐴𝐵) +𝑐 𝐴) ≈ (𝐴𝐵))
182, 16, 14, 17syl3anc 1475 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) +𝑐 𝐴) ≈ (𝐴𝐵))
19 domentr 8167 . . 3 (((𝐴 +𝑐 𝐵) ≼ ((𝐴𝐵) +𝑐 𝐴) ∧ ((𝐴𝐵) +𝑐 𝐴) ≈ (𝐴𝐵)) → (𝐴 +𝑐 𝐵) ≼ (𝐴𝐵))
2010, 18, 19syl2anc 565 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≼ (𝐴𝐵))
21 uncdadom 9194 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ≼ (𝐴 +𝑐 𝐵))
22213adant3 1125 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵) ≼ (𝐴 +𝑐 𝐵))
23 sbth 8235 . 2 (((𝐴 +𝑐 𝐵) ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ (𝐴 +𝑐 𝐵)) → (𝐴 +𝑐 𝐵) ≈ (𝐴𝐵))
2420, 22, 23syl2anc 565 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≈ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1070  wcel 2144  cun 3719  wss 3721   class class class wbr 4784  dom cdm 5249  (class class class)co 6792  ωcom 7211  cen 8105  cdom 8106  cardccrd 8960   +𝑐 ccda 9190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-oi 8570  df-card 8964  df-cda 9191
This theorem is referenced by:  alephadd  9600
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