Theorem etasslt 31920

Description: A restatement of noeta 31868 using set less than. (Contributed by Scott Fenton, 10-Dec-2021.)

Assertion

Ref	Expression
etasslt	⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem etasslt

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssltss1 31903	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
2		ssltex1 31901	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
3		ssltss2 31904	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
4		ssltex2 31902	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
5		ssltsep 31905	. . 3 ⊢ (𝐴 <<s 𝐵 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)
6		noeta 31868	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝑦 <s 𝑧) → ∃𝑥 ∈ No (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
7	1, 2, 3, 4, 5, 6	syl221anc 1337	. 2 ⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
8		brsslt 31900	. . . . . 6 ⊢ (𝐴 <<s {𝑥} ↔ ((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧)))
9		df-3an 1039	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ {𝑥} ⊆ No ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧) ↔ ((𝐴 ⊆ No ∧ {𝑥} ⊆ No ) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧))
10	9	bianass 842	. . . . . 6 ⊢ (((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧)) ↔ (((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No )) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧))
11	8, 10	bitri 264	. . . . 5 ⊢ (𝐴 <<s {𝑥} ↔ (((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No )) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧))
12	2	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → 𝐴 ∈ V)
13		snex 4908	. . . . . . . . . 10 ⊢ {𝑥} ∈ V
14	12, 13	jctir 561	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → (𝐴 ∈ V ∧ {𝑥} ∈ V))
15	1	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → 𝐴 ⊆ No )
16		snssi 4339	. . . . . . . . . 10 ⊢ (𝑥 ∈ No → {𝑥} ⊆ No )
17	16	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → {𝑥} ⊆ No )
18	14, 15, 17	jca32 558	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No )))
19	18	biantrurd 529	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ (((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No )) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧)))
20	19	bicomd 213	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ((((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No )) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧))
21		vex 3203	. . . . . . . 8 ⊢ 𝑥 ∈ V
22		breq2 4657	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝑥))
23	21, 22	ralsn 4222	. . . . . . 7 ⊢ (∀𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ 𝑦 <s 𝑥)
24	23	ralbii 2980	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ ∀𝑦 ∈ 𝐴 𝑦 <s 𝑥)
25	20, 24	syl6bb 276	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ((((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No )) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧) ↔ ∀𝑦 ∈ 𝐴 𝑦 <s 𝑥))
26	11, 25	syl5bb 272	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → (𝐴 <<s {𝑥} ↔ ∀𝑦 ∈ 𝐴 𝑦 <s 𝑥))
27		brsslt 31900	. . . . . . 7 ⊢ ({𝑥} <<s 𝐵 ↔ (({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)))
28		df-3an 1039	. . . . . . . 8 ⊢ (({𝑥} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧) ↔ (({𝑥} ⊆ No ∧ 𝐵 ⊆ No ) ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
29	28	bianass 842	. . . . . . 7 ⊢ ((({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)) ↔ ((({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No )) ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
30	27, 29	bitri 264	. . . . . 6 ⊢ ({𝑥} <<s 𝐵 ↔ ((({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No )) ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
31	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → 𝐵 ∈ V)
32	31, 13	jctil 560	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ({𝑥} ∈ V ∧ 𝐵 ∈ V))
33	3	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → 𝐵 ⊆ No )
34	32, 17, 33	jca32 558	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → (({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No )))
35	34	biantrurd 529	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → (∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧 ↔ ((({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No )) ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)))
36	35	bicomd 213	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → (((({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No )) ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧) ↔ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
37	30, 36	syl5bb 272	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ({𝑥} <<s 𝐵 ↔ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
38		breq1 4656	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 <s 𝑧 ↔ 𝑥 <s 𝑧))
39	38	ralbidv 2986	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝐵 𝑦 <s 𝑧 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧))
40	21, 39	ralsn 4222	. . . . 5 ⊢ (∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧)
41	37, 40	syl6bb 276	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ({𝑥} <<s 𝐵 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧))
42	26, 41	3anbi12d 1400	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) ↔ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))))
43	42	rexbidva 3049	. 2 ⊢ (𝐴 <<s 𝐵 → (∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) ↔ ∃𝑥 ∈ No (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))))
44	7, 43	mpbird 247	1 ⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  {csn 4177  ∪ cuni 4436   class class class wbr 4653   “ cima 5117  suc csuc 5725  ‘cfv 5888   No csur 31793   <s cslt 31794   bday cbday 31795   <<s csslt 31896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797  df-bday 31798  df-sslt 31897

This theorem is referenced by:  scutbdaybnd  31921