Theorem cdainflem 8896

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 8114	. . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω)
2		sdomnen 7870	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ≺ ω → ¬ (𝐴 ∪ 𝐵) ≈ ω)
3	1, 2	syl 17	. . 3 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → ¬ (𝐴 ∪ 𝐵) ≈ ω)
4	3	con2i 133	. 2 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → ¬ (𝐴 ≺ ω ∧ 𝐵 ≺ ω))
5		ianor 508	. . 3 ⊢ (¬ (𝐴 ≺ ω ∧ 𝐵 ≺ ω) ↔ (¬ 𝐴 ≺ ω ∨ ¬ 𝐵 ≺ ω))
6		relen 7846	. . . . . . . . . 10 ⊢ Rel ≈
7	6	brrelexi 5082	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (𝐴 ∪ 𝐵) ∈ V)
8		ssun1 3738	. . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
9		ssdomg 7887	. . . . . . . . 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 7901	. . . . . . . 8 ⊢ ((𝐴 ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≈ ω) → 𝐴 ≼ ω)
12	10, 11	mpancom 700	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → 𝐴 ≼ ω)
13	12	anim1i 590	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐴 ≺ ω) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
14		bren2 7872	. . . . . 6 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
15	13, 14	sylibr 223	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐴 ≺ ω) → 𝐴 ≈ ω)
16	15	ex 449	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (¬ 𝐴 ≺ ω → 𝐴 ≈ ω))
17		ssun2 3739	. . . . . . . . 9 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
18		ssdomg 7887	. . . . . . . . 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 7901	. . . . . . . 8 ⊢ ((𝐵 ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≈ ω) → 𝐵 ≼ ω)
21	19, 20	mpancom 700	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → 𝐵 ≼ ω)
22	21	anim1i 590	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐵 ≺ ω) → (𝐵 ≼ ω ∧ ¬ 𝐵 ≺ ω))
23		bren2 7872	. . . . . 6 ⊢ (𝐵 ≈ ω ↔ (𝐵 ≼ ω ∧ ¬ 𝐵 ≺ ω))
24	22, 23	sylibr 223	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐵 ≺ ω) → 𝐵 ≈ ω)
25	24	ex 449	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (¬ 𝐵 ≺ ω → 𝐵 ≈ ω))
26	16, 25	orim12d 879	. . 3 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → ((¬ 𝐴 ≺ ω ∨ ¬ 𝐵 ≺ ω) → (𝐴 ≈ ω ∨ 𝐵 ≈ ω)))
27	5, 26	syl5bi 231	. 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   ∨ wo 382   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540   class class class wbr 4583  ωcom 6957   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845

This theorem is referenced by:  cdainf  8897