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Theorem cdainflem 9602
Description: Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.)
Assertion
Ref Expression
cdainflem ((𝐴𝐵) ≈ ω → (𝐴 ≈ ω ∨ 𝐵 ≈ ω))

Proof of Theorem cdainflem
StepHypRef Expression
1 unfi2 8776 . . . 4 ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
2 sdomnen 8527 . . . 4 ((𝐴𝐵) ≺ ω → ¬ (𝐴𝐵) ≈ ω)
31, 2syl 17 . . 3 ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → ¬ (𝐴𝐵) ≈ ω)
43con2i 141 . 2 ((𝐴𝐵) ≈ ω → ¬ (𝐴 ≺ ω ∧ 𝐵 ≺ ω))
5 ianor 977 . . 3 (¬ (𝐴 ≺ ω ∧ 𝐵 ≺ ω) ↔ (¬ 𝐴 ≺ ω ∨ ¬ 𝐵 ≺ ω))
6 relen 8503 . . . . . . . . . 10 Rel ≈
76brrelex1i 5607 . . . . . . . . 9 ((𝐴𝐵) ≈ ω → (𝐴𝐵) ∈ V)
8 ssun1 4152 . . . . . . . . 9 𝐴 ⊆ (𝐴𝐵)
9 ssdomg 8544 . . . . . . . . 9 ((𝐴𝐵) ∈ V → (𝐴 ⊆ (𝐴𝐵) → 𝐴 ≼ (𝐴𝐵)))
107, 8, 9mpisyl 21 . . . . . . . 8 ((𝐴𝐵) ≈ ω → 𝐴 ≼ (𝐴𝐵))
11 domentr 8557 . . . . . . . 8 ((𝐴 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≈ ω) → 𝐴 ≼ ω)
1210, 11mpancom 684 . . . . . . 7 ((𝐴𝐵) ≈ ω → 𝐴 ≼ ω)
1312anim1i 614 . . . . . 6 (((𝐴𝐵) ≈ ω ∧ ¬ 𝐴 ≺ ω) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
14 bren2 8529 . . . . . 6 (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
1513, 14sylibr 235 . . . . 5 (((𝐴𝐵) ≈ ω ∧ ¬ 𝐴 ≺ ω) → 𝐴 ≈ ω)
1615ex 413 . . . 4 ((𝐴𝐵) ≈ ω → (¬ 𝐴 ≺ ω → 𝐴 ≈ ω))
17 ssun2 4153 . . . . . . . . 9 𝐵 ⊆ (𝐴𝐵)
18 ssdomg 8544 . . . . . . . . 9 ((𝐴𝐵) ∈ V → (𝐵 ⊆ (𝐴𝐵) → 𝐵 ≼ (𝐴𝐵)))
197, 17, 18mpisyl 21 . . . . . . . 8 ((𝐴𝐵) ≈ ω → 𝐵 ≼ (𝐴𝐵))
20 domentr 8557 . . . . . . . 8 ((𝐵 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≈ ω) → 𝐵 ≼ ω)
2119, 20mpancom 684 . . . . . . 7 ((𝐴𝐵) ≈ ω → 𝐵 ≼ ω)
2221anim1i 614 . . . . . 6 (((𝐴𝐵) ≈ ω ∧ ¬ 𝐵 ≺ ω) → (𝐵 ≼ ω ∧ ¬ 𝐵 ≺ ω))
23 bren2 8529 . . . . . 6 (𝐵 ≈ ω ↔ (𝐵 ≼ ω ∧ ¬ 𝐵 ≺ ω))
2422, 23sylibr 235 . . . . 5 (((𝐴𝐵) ≈ ω ∧ ¬ 𝐵 ≺ ω) → 𝐵 ≈ ω)
2524ex 413 . . . 4 ((𝐴𝐵) ≈ ω → (¬ 𝐵 ≺ ω → 𝐵 ≈ ω))
2616, 25orim12d 960 . . 3 ((𝐴𝐵) ≈ ω → ((¬ 𝐴 ≺ ω ∨ ¬ 𝐵 ≺ ω) → (𝐴 ≈ ω ∨ 𝐵 ≈ ω)))
275, 26syl5bi 243 . 2 ((𝐴𝐵) ≈ ω → (¬ (𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ≈ ω ∨ 𝐵 ≈ ω)))
284, 27mpd 15 1 ((𝐴𝐵) ≈ ω → (𝐴 ≈ ω ∨ 𝐵 ≈ ω))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 843  wcel 2107  Vcvv 3500  cun 3938  wss 3940   class class class wbr 5063  ωcom 7568  cen 8495  cdom 8496  csdm 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502
This theorem is referenced by:  djuinf  9603
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