Theorem right1s 28287

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 28241	. 2 ⊢ (R‘ 1s ) = {𝑥 ∈ (O‘(bday‘ 1s )) ∣ 1s <s 𝑥}
2		bday1 28205	. . . . . 6 ⊢ (bday‘ 1s ) = 1o
3	2	fveq2i 6888	. . . . 5 ⊢ (O‘(bday‘ 1s )) = (O‘1o)
4		old1 28256	. . . . 5 ⊢ (O‘1o) = { 0s }
5	3, 4	eqtri 2783	. . . 4 ⊢ (O‘(bday‘ 1s )) = { 0s }
6	5	rabeqi 3425	. . 3 ⊢ {𝑥 ∈ (O‘(bday‘ 1s )) ∣ 1s <s 𝑥} = {𝑥 ∈ { 0s } ∣ 1s <s 𝑥}
7		breq2 5106	. . . 4 ⊢ (𝑥 = 0s → ( 1s <s 𝑥 ↔ 1s <s 0s ))
8	7	rabsnif 4680	. . 3 ⊢ {𝑥 ∈ { 0s } ∣ 1s <s 𝑥} = if( 1s <s 0s , { 0s }, ∅)
9	6, 8	eqtri 2783	. 2 ⊢ {𝑥 ∈ (O‘(bday‘ 1s )) ∣ 1s <s 𝑥} = if( 1s <s 0s , { 0s }, ∅)
10		0lt1s 28203	. . . 4 ⊢ 0s <s 1s
11		0no 28200	. . . . 5 ⊢ 0s ∈ No
12		1no 28201	. . . . 5 ⊢ 1s ∈ No
13		ltsasym 28110	. . . . 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 4491	. 2 ⊢ if( 1s <s 0s , { 0s }, ∅) = ∅
17	1, 9, 16	3eqtri 2787	1 ⊢ (R‘ 1s ) = ∅

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2144  {crab 3412  ∅c0 4278  ifcif 4481  {csn 4584   class class class wbr 5102  ‘cfv 6538  1_oc1o 8472  Nocsur 28002   <s clts 28003  bdaycbday 28004   0_s c0s 28196   1_s c1s 28197  Ocold 28214  Rcright 28217

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-1s 28199  df-made 28218  df-old 28219  df-right 28222

This theorem is used by:  neg1s  28418  mulsrid  28504  1ons  28648  1reno  28888