Theorem tfinnnul 4490

Description: If M is a nonempty natural, then T_fin M is also nonempty. Corollary 1 of Theorem X.1.28 of [Rosser] p. 528. (Contributed by SF, 23-Jan-2015.)

Assertion

Ref	Expression
tfinnnul	|- ((M e. Nn /\ M =/= (/)) -> Tfin M =/= (/))

Proof of Theorem tfinnnul

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfinprop 4489	. 2 |- ((M e. Nn /\ M =/= (/)) -> ( Tfin M e. Nn /\ E.x e. M ~P1 x e. Tfin M))
2		ne0i 3556	. . . 4 |- (~P1 x e. Tfin M -> Tfin M =/= (/))
3	2	rexlimivw 2734	. . 3 |- (E.x e. M ~P1 x e. Tfin M -> Tfin M =/= (/))
4	3	adantl 452	. 2 |- (( Tfin M e. Nn /\ E.x e. M ~P1 x e. Tfin M) -> Tfin M =/= (/))
5	1, 4	syl 15	1 |- ((M e. Nn /\ M =/= (/)) -> Tfin M =/= (/))

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1710   =/= wne 2516  E.wrex 2615  (/)c0 3550  ~P₁ cpw1 4135   Nn cnnc 4373   T_fin ctfin 4435

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  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-rab 2623  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-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-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-tfin 4443

This theorem is referenced by:  tfin11  4493  tfinltfinlem1  4500  eventfin  4517  oddtfin  4518