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Theorem axtgcont 24067
Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 24066. (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.
StepHypRef 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 ) )
323expb 1195 . . . 4  |-  ( (
ph  /\  ( u  e.  S  /\  v  e.  T ) )  ->  u  e.  ( A I v ) )
43ralrimivva 2875 . . 3  |-  ( ph  ->  A. u  e.  S  A. v  e.  T  u  e.  ( A I v ) )
5 simplr 753 . . . . . . 7  |-  ( ( ( a  =  A  /\  x  =  u )  /\  y  =  v )  ->  x  =  u )
6 simpll 751 . . . . . . . 8  |-  ( ( ( a  =  A  /\  x  =  u )  /\  y  =  v )  ->  a  =  A )
7 simpr 459 . . . . . . . 8  |-  ( ( ( a  =  A  /\  x  =  u )  /\  y  =  v )  ->  y  =  v )
86, 7oveq12d 6288 . . . . . . 7  |-  ( ( ( a  =  A  /\  x  =  u )  /\  y  =  v )  ->  (
a I y )  =  ( A I v ) )
95, 8eleq12d 2536 . . . . . 6  |-  ( ( ( a  =  A  /\  x  =  u )  /\  y  =  v )  ->  (
x  e.  ( a I y )  <->  u  e.  ( A I v ) ) )
109cbvraldva 3087 . . . . 5  |-  ( ( a  =  A  /\  x  =  u )  ->  ( A. y  e.  T  x  e.  ( a I y )  <->  A. v  e.  T  u  e.  ( A I v ) ) )
1110cbvraldva 3087 . . . 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 ) ) )
1211rspcev 3207 . . 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 ) )
131, 4, 12syl2anc 659 . 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 )
2014, 15, 16, 17, 18, 19axtgcont1 24066 . 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 ) ) )
2113, 20mpd 15 1  |-  ( ph  ->  E. b  e.  P  A. x  e.  S  A. y  e.  T  b  e.  ( x I y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805    C_ wss 3461   ` cfv 5570  (class class class)co 6270   Basecbs 14719   distcds 14796  TarskiGcstrkg 24026  Itvcitv 24033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-trkgb 24046  df-trkg 24051
This theorem is referenced by:  f1otrg  24379
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