Theorem sucpw1nss3 7452

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 7448	. 2 |- ( -. EXMID -> -. ~P 1o e. 3o )
2		pw1on 7443	. . 3 |- ~P 1o e. On
3		sucssel 4521	. . 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 3200   ~Pcpw 3652  EXMIDwem 4284   Oncon0 4460   suc csuc 4462   1oc1o 6574   3oc3o 6576

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-tr 4188  df-exmid 4285  df-iord 4463  df-on 4465  df-suc 4468  df-1o 6581  df-2o 6582  df-3o 6583

This theorem is referenced by:  onntri45  7458