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Mirrors > Home > NFE Home > Th. List > vvex | GIF version |
Description: The universal class exists. This marks a major departure from ZFC set theory, where V is a proper class. (Contributed by SF, 12-Jan-2015.) |
Ref | Expression |
---|---|
vvex | ⊢ V ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncompl 4074 | . 2 ⊢ (x ∪ ∼ x) = V | |
2 | vex 2862 | . . 3 ⊢ x ∈ V | |
3 | 2 | complex 4104 | . . 3 ⊢ ∼ x ∈ V |
4 | 2, 3 | unex 4106 | . 2 ⊢ (x ∪ ∼ x) ∈ V |
5 | 1, 4 | eqeltrri 2424 | 1 ⊢ V ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1710 Vcvv 2859 ∼ ccompl 3205 ∪ cun 3207 |
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-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 |
This theorem is referenced by: 0ex 4110 cnvkexg 4286 uni1exg 4292 ssetkex 4294 imakexg 4299 pw1exg 4302 ins2kexg 4305 ins3kexg 4306 cokexg 4309 imagekexg 4311 abexv 4324 pwexg 4328 nnsucelrlem1 4424 preaddccan2lem1 4454 ltfinex 4464 ssfin 4470 ncfinraiselem2 4480 ncfinlowerlem1 4482 tfinrelkex 4487 evenfinex 4503 oddfinex 4504 evenodddisjlem1 4515 nnadjoinlem1 4519 srelkex 4525 tfinnnlem1 4533 vfinspnn 4541 1cvsfin 4542 tncveqnc1fin 4544 vfintle 4546 vfinncvntnn 4548 phiexg 4571 opexg 4587 proj1exg 4591 proj2exg 4592 setconslem5 4735 1stex 4739 swapex 4742 ssetex 4744 imaexg 4746 coexg 4749 siexg 4752 rnexg 5104 resexg 5116 ins2exg 5795 ins3exg 5796 mptexlem 5810 mpt2exlem 5811 cupex 5816 composeex 5820 addcfnex 5824 fnsex 5832 crossex 5850 domfnex 5870 ranfnex 5871 transex 5910 foundex 5914 ider 5943 ssetpov 5944 eqer 5961 ener 6039 enmap2lem1 6063 enmap1lem1 6069 enpw 6087 ncsex 6111 ncpw1c 6154 1p1e2c 6155 2p1e3c 6156 ce0addcnnul 6179 ce2 6192 ce2nc1 6193 lecncvg 6199 tcncv 6226 tcfnex 6244 ncvsq 6256 vvsqenvv 6257 csucex 6259 addccan2nclem2 6264 nmembers1lem1 6268 nncdiv3lem2 6276 nchoicelem11 6299 nchoicelem19 6307 fnfreclem1 6317 |
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