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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ctbnfien | Structured version Visualization version GIF version | ||
| Description: An infinite subset of a countable set is countable, without using choice. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) | 
| Ref | Expression | 
|---|---|
| ctbnfien | ⊢ (((𝑋 ≈ ω ∧ 𝑌 ≈ ω) ∧ (𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin)) → 𝐴 ≈ 𝑌) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isfinite 9692 | . . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω) | |
| 2 | 1 | notbii 320 | . . . 4 ⊢ (¬ 𝐴 ∈ Fin ↔ ¬ 𝐴 ≺ ω) | 
| 3 | relen 8990 | . . . . . . . . . . 11 ⊢ Rel ≈ | |
| 4 | 3 | brrelex1i 5741 | . . . . . . . . . 10 ⊢ (𝑋 ≈ ω → 𝑋 ∈ V) | 
| 5 | ssdomg 9040 | . . . . . . . . . 10 ⊢ (𝑋 ∈ V → (𝐴 ⊆ 𝑋 → 𝐴 ≼ 𝑋)) | |
| 6 | 4, 5 | syl 17 | . . . . . . . . 9 ⊢ (𝑋 ≈ ω → (𝐴 ⊆ 𝑋 → 𝐴 ≼ 𝑋)) | 
| 7 | domen2 9160 | . . . . . . . . 9 ⊢ (𝑋 ≈ ω → (𝐴 ≼ 𝑋 ↔ 𝐴 ≼ ω)) | |
| 8 | 6, 7 | sylibd 239 | . . . . . . . 8 ⊢ (𝑋 ≈ ω → (𝐴 ⊆ 𝑋 → 𝐴 ≼ ω)) | 
| 9 | 8 | imp 406 | . . . . . . 7 ⊢ ((𝑋 ≈ ω ∧ 𝐴 ⊆ 𝑋) → 𝐴 ≼ ω) | 
| 10 | brdom2 9022 | . . . . . . 7 ⊢ (𝐴 ≼ ω ↔ (𝐴 ≺ ω ∨ 𝐴 ≈ ω)) | |
| 11 | 9, 10 | sylib 218 | . . . . . 6 ⊢ ((𝑋 ≈ ω ∧ 𝐴 ⊆ 𝑋) → (𝐴 ≺ ω ∨ 𝐴 ≈ ω)) | 
| 12 | 11 | adantlr 715 | . . . . 5 ⊢ (((𝑋 ≈ ω ∧ 𝑌 ≈ ω) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ≺ ω ∨ 𝐴 ≈ ω)) | 
| 13 | 12 | ord 865 | . . . 4 ⊢ (((𝑋 ≈ ω ∧ 𝑌 ≈ ω) ∧ 𝐴 ⊆ 𝑋) → (¬ 𝐴 ≺ ω → 𝐴 ≈ ω)) | 
| 14 | 2, 13 | biimtrid 242 | . . 3 ⊢ (((𝑋 ≈ ω ∧ 𝑌 ≈ ω) ∧ 𝐴 ⊆ 𝑋) → (¬ 𝐴 ∈ Fin → 𝐴 ≈ ω)) | 
| 15 | 14 | impr 454 | . 2 ⊢ (((𝑋 ≈ ω ∧ 𝑌 ≈ ω) ∧ (𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin)) → 𝐴 ≈ ω) | 
| 16 | enen2 9158 | . . 3 ⊢ (𝑌 ≈ ω → (𝐴 ≈ 𝑌 ↔ 𝐴 ≈ ω)) | |
| 17 | 16 | ad2antlr 727 | . 2 ⊢ (((𝑋 ≈ ω ∧ 𝑌 ≈ ω) ∧ (𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin)) → (𝐴 ≈ 𝑌 ↔ 𝐴 ≈ ω)) | 
| 18 | 15, 17 | mpbird 257 | 1 ⊢ (((𝑋 ≈ ω ∧ 𝑌 ≈ ω) ∧ (𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin)) → 𝐴 ≈ 𝑌) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 ≺ csdm 8984 Fincfn 8985 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 | 
| This theorem is referenced by: fiphp3d 42830 irrapx1 42839 | 
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