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Mirrors > Home > NFE Home > Th. List > vfinnc | GIF version |
Description: If the universe is finite, then there is a unique natural containing any set. Theorem X.1.22 of [Rosser] p. 527. (Contributed by SF, 19-Jan-2015.) |
Ref | Expression |
---|---|
vfinnc | ⊢ ((A ∈ V ∧ V ∈ Fin ) → ∃!x ∈ Nn A ∈ x) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3291 | . . . 4 ⊢ A ⊆ V | |
2 | ssfin 4470 | . . . 4 ⊢ ((A ∈ V ∧ V ∈ Fin ∧ A ⊆ V) → A ∈ Fin ) | |
3 | 1, 2 | mp3an3 1266 | . . 3 ⊢ ((A ∈ V ∧ V ∈ Fin ) → A ∈ Fin ) |
4 | elfin 4420 | . . 3 ⊢ (A ∈ Fin ↔ ∃x ∈ Nn A ∈ x) | |
5 | 3, 4 | sylib 188 | . 2 ⊢ ((A ∈ V ∧ V ∈ Fin ) → ∃x ∈ Nn A ∈ x) |
6 | nnceleq 4430 | . . . . 5 ⊢ (((x ∈ Nn ∧ y ∈ Nn ) ∧ (A ∈ x ∧ A ∈ y)) → x = y) | |
7 | 6 | ex 423 | . . . 4 ⊢ ((x ∈ Nn ∧ y ∈ Nn ) → ((A ∈ x ∧ A ∈ y) → x = y)) |
8 | 7 | rgen2a 2680 | . . 3 ⊢ ∀x ∈ Nn ∀y ∈ Nn ((A ∈ x ∧ A ∈ y) → x = y) |
9 | 8 | a1i 10 | . 2 ⊢ ((A ∈ V ∧ V ∈ Fin ) → ∀x ∈ Nn ∀y ∈ Nn ((A ∈ x ∧ A ∈ y) → x = y)) |
10 | eleq2 2414 | . . 3 ⊢ (x = y → (A ∈ x ↔ A ∈ y)) | |
11 | 10 | reu4 3030 | . 2 ⊢ (∃!x ∈ Nn A ∈ x ↔ (∃x ∈ Nn A ∈ x ∧ ∀x ∈ Nn ∀y ∈ Nn ((A ∈ x ∧ A ∈ y) → x = y))) |
12 | 5, 9, 11 | sylanbrc 645 | 1 ⊢ ((A ∈ V ∧ V ∈ Fin ) → ∃!x ∈ Nn A ∈ x) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∈ wcel 1710 ∀wral 2614 ∃wrex 2615 ∃!wreu 2616 Vcvv 2859 ⊆ wss 3257 Nn cnnc 4373 Fin cfin 4376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 |
This theorem is referenced by: ncfinprop 4474 |
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