Theorem axtgcont 25368

Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 25367. (Contributed by Thierry Arnoux, 16-Mar-2019.)

Hypotheses

Ref	Expression
axtrkg.p	|- P = ( Base ` G )
axtrkg.d	|- .- = ( dist ` G )
axtrkg.i	|- I = (Itv ` G )
axtrkg.g	|- ( ph -> G e. TarskiG )
axtgcont.1	|- ( ph -> S C_ P )
axtgcont.2	|- ( ph -> T C_ P )
axtgcont.3	|- ( ph -> A e. P )
axtgcont.4	|- ( ( ph /\ u e. S /\ v e. T ) -> u e. ( A I v ) )

Assertion

Ref	Expression
axtgcont	|- ( ph -> E. b e. P A. x e. S A. y e. T b e. ( x I y ) )

Distinct variable groups:   x, y   v, b, A, u, x, y   I, b   v, u, x, y, I   P, b, u, v, x, y   S, b, x   T, b, x, y   .- , b, u, v, x, y   ph, u, v   u, S, v   u, T, v   u, A, x, y

Allowed substitution hints:   ph( x, y, b)   S( y)   G( x, y, v, u, b)

Proof of Theorem axtgcont

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axtgcont.3	. . 3 |- ( ph -> A e. P )
2		axtgcont.4	. . . . 5 |- ( ( ph /\ u e. S /\ v e. T ) -> u e. ( A I v ) )
3	2	3expb 1266	. . . 4 |- ( ( ph /\ ( u e. S /\ v e. T ) ) -> u e. ( A I v ) )
4	3	ralrimivva 2971	. . 3 |- ( ph -> A. u e. S A. v e. T u e. ( A I v ) )
5		simplr 792	. . . . . . 7 |- ( ( ( a = A /\ x = u ) /\ y = v ) -> x = u )
6		simpll 790	. . . . . . . 8 |- ( ( ( a = A /\ x = u ) /\ y = v ) -> a = A )
7		simpr 477	. . . . . . . 8 |- ( ( ( a = A /\ x = u ) /\ y = v ) -> y = v )
8	6, 7	oveq12d 6668	. . . . . . 7 |- ( ( ( a = A /\ x = u ) /\ y = v ) -> ( a I y ) = ( A I v ) )
9	5, 8	eleq12d 2695	. . . . . 6 |- ( ( ( a = A /\ x = u ) /\ y = v ) -> ( x e. ( a I y ) <-> u e. ( A I v ) ) )
10	9	cbvraldva 3177	. . . . 5 |- ( ( a = A /\ x = u ) -> ( A. y e. T x e. ( a I y ) <-> A. v e. T u e. ( A I v ) ) )
11	10	cbvraldva 3177	. . . 4 |- ( a = A -> ( A. x e. S A. y e. T x e. ( a I y ) <-> A. u e. S A. v e. T u e. ( A I v ) ) )
12	11	rspcev 3309	. . 3 |- ( ( A e. P /\ A. u e. S A. v e. T u e. ( A I v ) ) -> E. a e. P A. x e. S A. y e. T x e. ( a I y ) )
13	1, 4, 12	syl2anc 693	. 2 |- ( ph -> E. a e. P A. x e. S A. y e. T x e. ( a I y ) )
14		axtrkg.p	. . 3 |- P = ( Base ` G )
15		axtrkg.d	. . 3 |- .- = ( dist ` G )
16		axtrkg.i	. . 3 |- I = (Itv ` G )
17		axtrkg.g	. . 3 |- ( ph -> G e. TarskiG )
18		axtgcont.1	. . 3 |- ( ph -> S C_ P )
19		axtgcont.2	. . 3 |- ( ph -> T C_ P )
20	14, 15, 16, 17, 18, 19	axtgcont1 25367	. 2 |- ( ph -> ( E. a e. P A. x e. S A. y e. T x e. ( a I y ) -> E. b e. P A. x e. S A. y e. T b e. ( x I y ) ) )
21	13, 20	mpd 15	1 |- ( ph -> E. b e. P A. x e. S A. y e. T b e. ( x I y ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-trkgb 25348  df-trkg 25352

This theorem is referenced by:  f1otrg  25751