MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cdainflem Structured version   Visualization version   GIF version

Theorem cdainflem 9687
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 8861 . . . 4 ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
2 sdomnen 8584 . . . 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 981 . . 3 (¬ (𝐴 ≺ ω ∧ 𝐵 ≺ ω) ↔ (¬ 𝐴 ≺ ω ∨ ¬ 𝐵 ≺ ω))
6 relen 8560 . . . . . . . . . 10 Rel ≈
76brrelex1i 5579 . . . . . . . . 9 ((𝐴𝐵) ≈ ω → (𝐴𝐵) ∈ V)
8 ssun1 4062 . . . . . . . . 9 𝐴 ⊆ (𝐴𝐵)
9 ssdomg 8601 . . . . . . . . 9 ((𝐴𝐵) ∈ V → (𝐴 ⊆ (𝐴𝐵) → 𝐴 ≼ (𝐴𝐵)))
107, 8, 9mpisyl 21 . . . . . . . 8 ((𝐴𝐵) ≈ ω → 𝐴 ≼ (𝐴𝐵))
11 domentr 8614 . . . . . . . 8 ((𝐴 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≈ ω) → 𝐴 ≼ ω)
1210, 11mpancom 688 . . . . . . 7 ((𝐴𝐵) ≈ ω → 𝐴 ≼ ω)
1312anim1i 618 . . . . . 6 (((𝐴𝐵) ≈ ω ∧ ¬ 𝐴 ≺ ω) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
14 bren2 8586 . . . . . 6 (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
1513, 14sylibr 237 . . . . 5 (((𝐴𝐵) ≈ ω ∧ ¬ 𝐴 ≺ ω) → 𝐴 ≈ ω)
1615ex 416 . . . 4 ((𝐴𝐵) ≈ ω → (¬ 𝐴 ≺ ω → 𝐴 ≈ ω))
17 ssun2 4063 . . . . . . . . 9 𝐵 ⊆ (𝐴𝐵)
18 ssdomg 8601 . . . . . . . . 9 ((𝐴𝐵) ∈ V → (𝐵 ⊆ (𝐴𝐵) → 𝐵 ≼ (𝐴𝐵)))
197, 17, 18mpisyl 21 . . . . . . . 8 ((𝐴𝐵) ≈ ω → 𝐵 ≼ (𝐴𝐵))
20 domentr 8614 . . . . . . . 8 ((𝐵 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≈ ω) → 𝐵 ≼ ω)
2119, 20mpancom 688 . . . . . . 7 ((𝐴𝐵) ≈ ω → 𝐵 ≼ ω)
2221anim1i 618 . . . . . 6 (((𝐴𝐵) ≈ ω ∧ ¬ 𝐵 ≺ ω) → (𝐵 ≼ ω ∧ ¬ 𝐵 ≺ ω))
23 bren2 8586 . . . . . 6 (𝐵 ≈ ω ↔ (𝐵 ≼ ω ∧ ¬ 𝐵 ≺ ω))
2422, 23sylibr 237 . . . . 5 (((𝐴𝐵) ≈ ω ∧ ¬ 𝐵 ≺ ω) → 𝐵 ≈ ω)
2524ex 416 . . . 4 ((𝐴𝐵) ≈ ω → (¬ 𝐵 ≺ ω → 𝐵 ≈ ω))
2616, 25orim12d 964 . . 3 ((𝐴𝐵) ≈ ω → ((¬ 𝐴 ≺ ω ∨ ¬ 𝐵 ≺ ω) → (𝐴 ≈ ω ∨ 𝐵 ≈ ω)))
275, 26syl5bi 245 . 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 399  wo 846  wcel 2114  Vcvv 3398  cun 3841  wss 3843   class class class wbr 5030  ωcom 7599  cen 8552  cdom 8553  csdm 8554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7600  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559
This theorem is referenced by:  djuinf  9688
  Copyright terms: Public domain W3C validator