Theorem nfel 2992

Description: Hypothesis builder for elementhood. (Contributed by NM, 1-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)

Hypotheses

Ref	Expression
nfnfc.1	⊢ Ⅎ𝑥𝐴
nfeq.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfel	⊢ Ⅎ𝑥 𝐴 ∈ 𝐵

Proof of Theorem nfel

Step	Hyp	Ref	Expression
1		nfnfc.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
3		nfeq.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	3	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐵)
5	2, 4	nfeld 2989	. 2 ⊢ (⊤ → Ⅎ𝑥 𝐴 ∈ 𝐵)
6	5	mptru 1540	1 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵

Syntax hints:  ⊤wtru 1534  Ⅎwnf 1780   ∈ wcel 2110  Ⅎwnfc 2961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-cleq 2814  df-clel 2893  df-nfc 2963

This theorem is referenced by:  nfel1  2994  nfel2  2996  nfnel  3130  elabgf  3663  elrabf  3675  sbcel12  4359  rabxfrd  5317  ffnfvf  6882  nfixpw  8479  mptelixpg  8498  ptcldmpt  22221  prdsdsf  22976  prdsxmet  22978  ssiun2sf  30310  acunirnmpt2  30404  acunirnmpt2f  30405  aciunf1lem  30406  funcnv4mpt  30413  fsumiunle  30545  esumc  31310  esumrnmpt2  31327  esumgect  31349  esum2dlem  31351  esum2d  31352  esumiun  31353  ldsysgenld  31419  sigapildsys  31421  fiunelros  31433  omssubadd  31558  breprexplema  31901  bnj1491  32329  currysetlem  34256  currysetlem1  34258  ptrest  34890  aomclem8  39659  ss2iundf  40002  elunif  41271  rspcegf  41278  fiiuncl  41325  eliuniincex  41373  disjf1  41441  disjf1o  41450  disjinfi  41452  iunmapsn  41478  fmptf  41507  infnsuprnmpt  41520  iuneqfzuzlem  41600  allbutfi  41663  supminfrnmpt  41717  supminfxrrnmpt  41745  monoordxr  41757  monoord2xr  41759  iooiinicc  41816  iooiinioc  41830  fsumsplit1  41851  fsumiunss  41854  fprodcnlem  41878  fprodcn  41879  climsuse  41887  climsubmpt  41939  climreclf  41943  fnlimcnv  41946  climeldmeqmpt  41947  climfveqmpt  41950  fnlimfvre  41953  fnlimabslt  41958  climfveqmpt3  41961  climbddf  41966  climeldmeqmpt3  41968  climinf2mpt  41993  climinfmpt  41994  limsupequzmptf  42010  lmbr3  42026  fprodcncf  42182  dvmptmulf  42220  dvnmptdivc  42221  dvnmul  42226  dvmptfprodlem  42227  dvmptfprod  42228  dvnprodlem1  42229  dvnprodlem2  42230  stoweidlem59  42343  fourierdlem31  42422  sge00  42657  sge0f1o  42663  sge0pnffigt  42677  sge0lefi  42679  sge0resplit  42687  sge0lempt  42691  sge0iunmptlemfi  42694  sge0iunmptlemre  42696  sge0iunmpt  42699  sge0xadd  42716  sge0gtfsumgt  42724  iundjiun  42741  meadjiun  42747  meaiininclem  42767  omeiunltfirp  42800  hoidmvlelem1  42876  hoidmvlelem3  42878  hspdifhsp  42897  hoiqssbllem2  42904  hspmbllem2  42908  opnvonmbllem2  42914  hoimbl2  42946  vonhoire  42953  iinhoiicc  42955  iunhoiioo  42957  vonn0ioo2  42971  vonn0icc2  42973  incsmflem  43017  issmfle  43021  issmfgt  43032  decsmflem  43041  issmfge  43045  smflimlem2  43047  smflim  43052  smfresal  43062  smfpimbor1lem2  43073  smflim2  43079  smflimmpt  43083  smfsuplem1  43084  smfsupmpt  43088  smfsupxr  43089  smfinflem  43090  smfinfmpt  43092  smflimsuplem7  43099  smflimsuplem8  43100  smflimsup  43101  smflimsupmpt  43102  smfliminf  43104  smfliminfmpt  43105  nfdfat  43325