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Mirrors > Home > MPE Home > Th. List > cdainflem | Structured version Visualization version GIF version |
Description: Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.) |
Ref | Expression |
---|---|
cdainflem | ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (𝐴 ≈ ω ∨ 𝐵 ≈ ω)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unfi2 8775 | . . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω) | |
2 | sdomnen 8526 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ≺ ω → ¬ (𝐴 ∪ 𝐵) ≈ ω) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → ¬ (𝐴 ∪ 𝐵) ≈ ω) |
4 | 3 | con2i 141 | . 2 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → ¬ (𝐴 ≺ ω ∧ 𝐵 ≺ ω)) |
5 | ianor 975 | . . 3 ⊢ (¬ (𝐴 ≺ ω ∧ 𝐵 ≺ ω) ↔ (¬ 𝐴 ≺ ω ∨ ¬ 𝐵 ≺ ω)) | |
6 | relen 8502 | . . . . . . . . . 10 ⊢ Rel ≈ | |
7 | 6 | brrelex1i 5601 | . . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (𝐴 ∪ 𝐵) ∈ V) |
8 | ssun1 4145 | . . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
9 | ssdomg 8543 | . . . . . . . . 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 8556 | . . . . . . . 8 ⊢ ((𝐴 ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≈ ω) → 𝐴 ≼ ω) | |
12 | 10, 11 | mpancom 684 | . . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → 𝐴 ≼ ω) |
13 | 12 | anim1i 614 | . . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐴 ≺ ω) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω)) |
14 | bren2 8528 | . . . . . 6 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω)) | |
15 | 13, 14 | sylibr 235 | . . . . 5 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐴 ≺ ω) → 𝐴 ≈ ω) |
16 | 15 | ex 413 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (¬ 𝐴 ≺ ω → 𝐴 ≈ ω)) |
17 | ssun2 4146 | . . . . . . . . 9 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
18 | ssdomg 8543 | . . . . . . . . 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 8556 | . . . . . . . 8 ⊢ ((𝐵 ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≈ ω) → 𝐵 ≼ ω) | |
21 | 19, 20 | mpancom 684 | . . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → 𝐵 ≼ ω) |
22 | 21 | anim1i 614 | . . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐵 ≺ ω) → (𝐵 ≼ ω ∧ ¬ 𝐵 ≺ ω)) |
23 | bren2 8528 | . . . . . 6 ⊢ (𝐵 ≈ ω ↔ (𝐵 ≼ ω ∧ ¬ 𝐵 ≺ ω)) | |
24 | 22, 23 | sylibr 235 | . . . . 5 ⊢ (((𝐴 ∪ 𝐵) ≈ ω ∧ ¬ 𝐵 ≺ ω) → 𝐵 ≈ ω) |
25 | 24 | ex 413 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (¬ 𝐵 ≺ ω → 𝐵 ≈ ω)) |
26 | 16, 25 | orim12d 958 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ≈ ω → ((¬ 𝐴 ≺ ω ∨ ¬ 𝐵 ≺ ω) → (𝐴 ≈ ω ∨ 𝐵 ≈ ω))) |
27 | 5, 26 | syl5bi 243 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 841 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ⊆ wss 3933 class class class wbr 5057 ωcom 7569 ≈ cen 8494 ≼ cdom 8495 ≺ csdm 8496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 |
This theorem is referenced by: djuinf 9602 |
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