Theorem cdainflem 10090

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	⊢ ((𝐴 ∪ 𝐵) ≈ ω → (𝐴 ≈ ω ∨ 𝐵 ≈ ω))

Proof of Theorem cdainflem

Step	Hyp	Ref	Expression
1		unfi2 9205	. . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω)
2		sdomnen 8914	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ≺ ω → ¬ (𝐴 ∪ 𝐵) ≈ ω)
3	1, 2	syl 17	. . 3 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → ¬ (𝐴 ∪ 𝐵) ≈ ω)
4	3	con2i 139	. 2 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → ¬ (𝐴 ≺ ω ∧ 𝐵 ≺ ω))
5		ianor 983	. . 3 ⊢ (¬ (𝐴 ≺ ω ∧ 𝐵 ≺ ω) ↔ (¬ 𝐴 ≺ ω ∨ ¬ 𝐵 ≺ ω))
6		relen 8884	. . . . . . . . . 10 ⊢ Rel ≈
7	6	brrelex1i 5677	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (𝐴 ∪ 𝐵) ∈ V)
8		ssun1 4127	. . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
9		ssdomg 8933	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ⊆ (𝐴 ∪ 𝐵) → 𝐴 ≼ (𝐴 ∪ 𝐵)))
10	7, 8, 9	mpisyl 21	. . . . . . . 8 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → 𝐴 ≼ (𝐴 ∪ 𝐵))
11		domentr 8946	. . . . . . . 8 ⊢ ((𝐴 ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≈ ω) → 𝐴 ≼ ω)
12	10, 11	mpancom 688	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → 𝐴 ≼ ω)
13	12	anim1i 615	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐴 ≺ ω) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
14		bren2 8916	. . . . . 6 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
15	13, 14	sylibr 234	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐴 ≺ ω) → 𝐴 ≈ ω)
16	15	ex 412	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (¬ 𝐴 ≺ ω → 𝐴 ≈ ω))
17		ssun2 4128	. . . . . . . . 9 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
18		ssdomg 8933	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐵 ⊆ (𝐴 ∪ 𝐵) → 𝐵 ≼ (𝐴 ∪ 𝐵)))
19	7, 17, 18	mpisyl 21	. . . . . . . 8 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → 𝐵 ≼ (𝐴 ∪ 𝐵))
20		domentr 8946	. . . . . . . 8 ⊢ ((𝐵 ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≈ ω) → 𝐵 ≼ ω)
21	19, 20	mpancom 688	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → 𝐵 ≼ ω)
22	21	anim1i 615	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐵 ≺ ω) → (𝐵 ≼ ω ∧ ¬ 𝐵 ≺ ω))
23		bren2 8916	. . . . . 6 ⊢ (𝐵 ≈ ω ↔ (𝐵 ≼ ω ∧ ¬ 𝐵 ≺ ω))
24	22, 23	sylibr 234	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐵 ≺ ω) → 𝐵 ≈ ω)
25	24	ex 412	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (¬ 𝐵 ≺ ω → 𝐵 ≈ ω))
26	16, 25	orim12d 966	. . 3 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → ((¬ 𝐴 ≺ ω ∨ ¬ 𝐵 ≺ ω) → (𝐴 ≈ ω ∨ 𝐵 ≈ ω)))
27	5, 26	biimtrid 242	. 2 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (¬ (𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ≈ ω ∨ 𝐵 ≈ ω)))
28	4, 27	mpd 15	1 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (𝐴 ≈ ω ∨ 𝐵 ≈ ω))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∈ wcel 2113  Vcvv 3437   ∪ cun 3896   ⊆ wss 3898   class class class wbr 5095  ωcom 7805   ≈ cen 8876   ≼ cdom 8877   ≺ csdm 8878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883

This theorem is referenced by:  djuinf  10091