Theorem etasslt 33276

Description: A restatement of noeta 33224 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 33259	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
2		ssltex1 33257	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
3		ssltss2 33260	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
4		ssltex2 33258	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
5		ssltsep 33261	. . 3 ⊢ (𝐴 <<s 𝐵 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)
6		noeta 33224	. . 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 1377	. 2 ⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
8		brsslt 33256	. . . . . 6 ⊢ (𝐴 <<s {𝑥} ↔ ((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧)))
9		df-3an 1085	. . . . . . 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 640	. . . . . 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 277	. . . . 5 ⊢ (𝐴 <<s {𝑥} ↔ (((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No )) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧))
12	2	adantr 483	. . . . . . . . . 10 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → 𝐴 ∈ V)
13		snex 5334	. . . . . . . . . 10 ⊢ {𝑥} ∈ V
14	12, 13	jctir 523	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → (𝐴 ∈ V ∧ {𝑥} ∈ V))
15	1	adantr 483	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → 𝐴 ⊆ No )
16		snssi 4743	. . . . . . . . . 10 ⊢ (𝑥 ∈ No → {𝑥} ⊆ No )
17	16	adantl 484	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → {𝑥} ⊆ No )
18	14, 15, 17	jca32 518	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No )))
19	18	biantrurd 535	. . . . . . 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 225	. . . . . 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 3499	. . . . . . . 8 ⊢ 𝑥 ∈ V
22		breq2 5072	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝑥))
23	21, 22	ralsn 4621	. . . . . . 7 ⊢ (∀𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ 𝑦 <s 𝑥)
24	23	ralbii 3167	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ ∀𝑦 ∈ 𝐴 𝑦 <s 𝑥)
25	20, 24	syl6bb 289	. . . . 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 285	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → (𝐴 <<s {𝑥} ↔ ∀𝑦 ∈ 𝐴 𝑦 <s 𝑥))
27		brsslt 33256	. . . . . . 7 ⊢ ({𝑥} <<s 𝐵 ↔ (({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)))
28		df-3an 1085	. . . . . . . 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 640	. . . . . . 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 277	. . . . . 6 ⊢ ({𝑥} <<s 𝐵 ↔ ((({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No )) ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
31	4	adantr 483	. . . . . . . . . 10 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → 𝐵 ∈ V)
32	31, 13	jctil 522	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ({𝑥} ∈ V ∧ 𝐵 ∈ V))
33	3	adantr 483	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → 𝐵 ⊆ No )
34	32, 17, 33	jca32 518	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → (({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No )))
35	34	biantrurd 535	. . . . . . 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 225	. . . . . 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 285	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ({𝑥} <<s 𝐵 ↔ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
38		breq1 5071	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 <s 𝑧 ↔ 𝑥 <s 𝑧))
39	38	ralbidv 3199	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝐵 𝑦 <s 𝑧 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧))
40	21, 39	ralsn 4621	. . . . 5 ⊢ (∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧)
41	37, 40	syl6bb 289	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ({𝑥} <<s 𝐵 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧))
42	26, 41	3anbi12d 1433	. . 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 3298	. 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 259	1 ⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ∪ cun 3936   ⊆ wss 3938  {csn 4569  ∪ cuni 4840   class class class wbr 5068   “ cima 5560  suc csuc 6195  ‘cfv 6357   No csur 33149   <s cslt 33150   bday cbday 33151   <<s csslt 33252

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-1o 8104  df-2o 8105  df-no 33152  df-slt 33153  df-bday 33154  df-sslt 33253

This theorem is referenced by:  scutbdaybnd  33277