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Mirrors > Home > MPE Home > Th. List > fineqv | Structured version Visualization version GIF version |
Description: If the Axiom of Infinity is denied, then all sets are finite (which implies the Axiom of Choice). (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 3-Jan-2015.) |
Ref | Expression |
---|---|
fineqv | ⊢ (¬ ω ∈ V ↔ Fin = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3625 | . . . 4 ⊢ Fin ⊆ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (¬ ω ∈ V → Fin ⊆ V) |
3 | vex 3203 | . . . . . . . 8 ⊢ 𝑎 ∈ V | |
4 | fineqvlem 8174 | . . . . . . . 8 ⊢ ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ≼ 𝒫 𝒫 𝑎) | |
5 | 3, 4 | mpan 706 | . . . . . . 7 ⊢ (¬ 𝑎 ∈ Fin → ω ≼ 𝒫 𝒫 𝑎) |
6 | reldom 7961 | . . . . . . . 8 ⊢ Rel ≼ | |
7 | 6 | brrelexi 5158 | . . . . . . 7 ⊢ (ω ≼ 𝒫 𝒫 𝑎 → ω ∈ V) |
8 | 5, 7 | syl 17 | . . . . . 6 ⊢ (¬ 𝑎 ∈ Fin → ω ∈ V) |
9 | 8 | con1i 144 | . . . . 5 ⊢ (¬ ω ∈ V → 𝑎 ∈ Fin) |
10 | 9 | a1d 25 | . . . 4 ⊢ (¬ ω ∈ V → (𝑎 ∈ V → 𝑎 ∈ Fin)) |
11 | 10 | ssrdv 3609 | . . 3 ⊢ (¬ ω ∈ V → V ⊆ Fin) |
12 | 2, 11 | eqssd 3620 | . 2 ⊢ (¬ ω ∈ V → Fin = V) |
13 | ominf 8172 | . . 3 ⊢ ¬ ω ∈ Fin | |
14 | eleq2 2690 | . . 3 ⊢ (Fin = V → (ω ∈ Fin ↔ ω ∈ V)) | |
15 | 13, 14 | mtbii 316 | . 2 ⊢ (Fin = V → ¬ ω ∈ V) |
16 | 12, 15 | impbii 199 | 1 ⊢ (¬ ω ∈ V ↔ Fin = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 𝒫 cpw 4158 class class class wbr 4653 ωcom 7065 ≼ cdom 7953 Fincfn 7955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 |
This theorem is referenced by: npomex 9818 |
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