Theorem sucpw1nss3 7347

Description: Negated excluded middle implies that the successor of the power set of 1o is not a subset of 3o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)

Assertion

Ref	Expression
sucpw1nss3	|- ( -. EXMID -> -. suc ~P 1o C_ 3o )

Proof of Theorem sucpw1nss3

Step	Hyp	Ref	Expression
1		pw1nel3 7343	. 2 |- ( -. EXMID -> -. ~P 1o e. 3o )
2		pw1on 7338	. . 3 |- ~P 1o e. On
3		sucssel 4471	. . 3 |- ( ~P 1o e. On -> ( suc ~P 1o C_ 3o -> ~P 1o e. 3o ) )
4	2, 3	ax-mp 5	. 2 |- ( suc ~P 1o C_ 3o -> ~P 1o e. 3o )
5	1, 4	nsyl 629	1 |- ( -. EXMID -> -. suc ~P 1o C_ 3o )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2176   C_ wss 3166   ~Pcpw 3616  EXMIDwem 4238   Oncon0 4410   suc csuc 4412   1oc1o 6495   3oc3o 6497

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-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-tr 4143  df-exmid 4239  df-iord 4413  df-on 4415  df-suc 4418  df-1o 6502  df-2o 6503  df-3o 6504

This theorem is referenced by:  onntri45  7353