Theorem xnn0xr 9383

Description: An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)

Assertion

Ref	Expression
xnn0xr	|- ( A e. NN0* -> A e. RR* )

Proof of Theorem xnn0xr

Step	Hyp	Ref	Expression
1		elxnn0 9380	. 2 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		nn0re 9324	. . . 4 |- ( A e. NN0 -> A e. RR )
3	2	rexrd 8142	. . 3 |- ( A e. NN0 -> A e. RR* )
4		pnfxr 8145	. . . 4 |- +oo e. RR*
5		eleq1 2269	. . . 4 |- ( A = +oo -> ( A e. RR* <-> +oo e. RR* ) )
6	4, 5	mpbiri 168	. . 3 |- ( A = +oo -> A e. RR* )
7	3, 6	jaoi 718	. 2 |- ( ( A e. NN0 \/ A = +oo ) -> A e. RR* )
8	1, 7	sylbi 121	1 |- ( A e. NN0* -> A e. RR* )

Syntax hints:   -> wi 4   \/ wo 710   = wceq 1373   e. wcel 2177   +oocpnf 8124   RR*cxr 8126   NN0cn0 9315  NN0*cxnn0 9378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-un 4488  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042  ax-rnegex 8054

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-uni 3857  df-int 3892  df-pnf 8129  df-xr 8131  df-inn 9057  df-n0 9316  df-xnn0 9379

This theorem is referenced by:  xnn0xrnemnf  9390  xnn0dcle  9944  xnn0letri  9945