Theorem sucpw1nss3 7496

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 7492	. 2 |- ( -. EXMID -> -. ~P 1o e. 3o )
2		pw1on 7487	. . 3 |- ~P 1o e. On
3		sucssel 4527	. . 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 633	1 |- ( -. EXMID -> -. suc ~P 1o C_ 3o )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2202   C_ wss 3201   ~Pcpw 3656  EXMIDwem 4290   Oncon0 4466   suc csuc 4468   1oc1o 6618   3oc3o 6620

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-tr 4193  df-exmid 4291  df-iord 4469  df-on 4471  df-suc 4474  df-1o 6625  df-2o 6626  df-3o 6627

This theorem is referenced by:  onntri45  7502