Theorem cdainflem 10257

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 9376	. . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω)
2		sdomnen 9041	. . . 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 982	. . 3 ⊢ (¬ (𝐴 ≺ ω ∧ 𝐵 ≺ ω) ↔ (¬ 𝐴 ≺ ω ∨ ¬ 𝐵 ≺ ω))
6		relen 9008	. . . . . . . . . 10 ⊢ Rel ≈
7	6	brrelex1i 5756	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (𝐴 ∪ 𝐵) ∈ V)
8		ssun1 4201	. . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
9		ssdomg 9060	. . . . . . . . 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 9073	. . . . . . . 8 ⊢ ((𝐴 ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≈ ω) → 𝐴 ≼ ω)
12	10, 11	mpancom 687	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → 𝐴 ≼ ω)
13	12	anim1i 614	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐴 ≺ ω) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
14		bren2 9043	. . . . . 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 4202	. . . . . . . . 9 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
18		ssdomg 9060	. . . . . . . . 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 9073	. . . . . . . 8 ⊢ ((𝐵 ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≈ ω) → 𝐵 ≼ ω)
21	19, 20	mpancom 687	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → 𝐵 ≼ ω)
22	21	anim1i 614	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐵 ≺ ω) → (𝐵 ≼ ω ∧ ¬ 𝐵 ≺ ω))
23		bren2 9043	. . . . . 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 965	. . 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 846   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976   class class class wbr 5166  ωcom 7903   ≈ cen 9000   ≼ cdom 9001   ≺ csdm 9002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007

This theorem is referenced by:  djuinf  10258