Theorem right1s 28315

Description: The right set of 1_s is empty . (Contributed by Scott Fenton, 4-Feb-2025.)

Assertion

Ref	Expression
right1s	⊢ (R‘ 1s ) = ∅

Proof of Theorem right1s

Step	Hyp	Ref	Expression
1		rightval 28269	. 2 ⊢ (R‘ 1s ) = {𝑥 ∈ (O‘(bday‘ 1s )) ∣ 1s <s 𝑥}
2		bday1 28233	. . . . . 6 ⊢ (bday‘ 1s ) = 1o
3	2	fveq2i 6914	. . . . 5 ⊢ (O‘(bday‘ 1s )) = (O‘1o)
4		old1 28284	. . . . 5 ⊢ (O‘1o) = { 0s }
5	3, 4	eqtri 2786	. . . 4 ⊢ (O‘(bday‘ 1s )) = { 0s }
6	5	rabeqi 3429	. . 3 ⊢ {𝑥 ∈ (O‘(bday‘ 1s )) ∣ 1s <s 𝑥} = {𝑥 ∈ { 0s } ∣ 1s <s 𝑥}
7		breq2 5110	. . . 4 ⊢ (𝑥 = 0s → ( 1s <s 𝑥 ↔ 1s <s 0s ))
8	7	rabsnif 4684	. . 3 ⊢ {𝑥 ∈ { 0s } ∣ 1s <s 𝑥} = if( 1s <s 0s , { 0s }, ∅)
9	6, 8	eqtri 2786	. 2 ⊢ {𝑥 ∈ (O‘(bday‘ 1s )) ∣ 1s <s 𝑥} = if( 1s <s 0s , { 0s }, ∅)
10		0lt1s 28231	. . . 4 ⊢ 0s <s 1s
11		0no 28228	. . . . 5 ⊢ 0s ∈ No
12		1no 28229	. . . . 5 ⊢ 1s ∈ No
13		ltsasym 28138	. . . . 5 ⊢ (( 0s ∈ No ∧ 1s ∈ No) → ( 0s <s 1s → ¬ 1s <s 0s ))
14	11, 12, 13	mp2an 705	. . . 4 ⊢ ( 0s <s 1s → ¬ 1s <s 0s )
15	10, 14	ax-mp 5	. . 3 ⊢ ¬ 1s <s 0s
16	15	iffalsei 4495	. 2 ⊢ if( 1s <s 0s , { 0s }, ∅) = ∅
17	1, 9, 16	3eqtri 2790	1 ⊢ (R‘ 1s ) = ∅

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2147  {crab 3416  ∅c0 4281  ifcif 4485  {csn 4588   class class class wbr 5106  ‘cfv 6561  1_oc1o 8500  Nocsur 28030   <s clts 28031  bdaycbday 28032   0_s c0s 28224   1_s c1s 28225  Ocold 28242  Rcright 28245

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-0s 28226  df-1s 28227  df-made 28246  df-old 28247  df-right 28250

This theorem is used by:  neg1s  28446  mulsrid  28532  1ons  28676  1reno  28916