Theorem negsproplem5 27974

Description: Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐵 is simpler than 𝐴. (Contributed by Scott Fenton, 3-Feb-2025.)

Hypotheses

Ref	Expression
negsproplem.1	⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
negsproplem4.1	⊢ (𝜑 → 𝐴 ∈ No )
negsproplem4.2	⊢ (𝜑 → 𝐵 ∈ No )
negsproplem4.3	⊢ (𝜑 → 𝐴 <s 𝐵)
negsproplem5.4	⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴))

Assertion

Ref	Expression
negsproplem5	⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem negsproplem5

Step	Hyp	Ref	Expression
1		negsproplem.1	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
2		negsproplem4.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ No )
3	1, 2	negsproplem3 27972	. . 3 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
4	3	simp2d 1143	. 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
5		negsfn 27965	. . 3 ⊢ -us Fn No
6		rightssno 27827	. . 3 ⊢ ( R ‘𝐴) ⊆ No
7		negsproplem5.4	. . . . 5 ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴))
8		bdayelon 27715	. . . . . 6 ⊢ ( bday ‘𝐴) ∈ On
9		negsproplem4.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
10		oldbday 27846	. . . . . 6 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
11	8, 9, 10	sylancr 587	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
12	7, 11	mpbird 257	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ( O ‘( bday ‘𝐴)))
13		negsproplem4.3	. . . 4 ⊢ (𝜑 → 𝐴 <s 𝐵)
14		elright 27807	. . . 4 ⊢ (𝐵 ∈ ( R ‘𝐴) ↔ (𝐵 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝐵))
15	12, 13, 14	sylanbrc 583	. . 3 ⊢ (𝜑 → 𝐵 ∈ ( R ‘𝐴))
16		fnfvima 7167	. . 3 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ∧ 𝐵 ∈ ( R ‘𝐴)) → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴)))
17	5, 6, 15, 16	mp3an12i 1467	. 2 ⊢ (𝜑 → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴)))
18		fvex 6835	. . . 4 ⊢ ( -us ‘𝐴) ∈ V
19	18	snid 4612	. . 3 ⊢ ( -us ‘𝐴) ∈ {( -us ‘𝐴)}
20	19	a1i 11	. 2 ⊢ (𝜑 → ( -us ‘𝐴) ∈ {( -us ‘𝐴)})
21	4, 17, 20	ssltsepcd 27735	1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2111  ∀wral 3047   ∪ cun 3895   ⊆ wss 3897  {csn 4573   class class class wbr 5089   “ cima 5617  Oncon0 6306   Fn wfn 6476  ‘cfv 6481   No csur 27578   <s cslt 27579   bday cbday 27580   <<s csslt 27720   O cold 27784   L cleft 27786   R cright 27787   -u_s cnegs 27961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-2o 8386  df-no 27581  df-slt 27582  df-bday 27583  df-sslt 27721  df-scut 27723  df-0s 27768  df-made 27788  df-old 27789  df-left 27791  df-right 27792  df-norec 27881  df-negs 27963

This theorem is referenced by:  negsproplem7  27976