Theorem etasslt 32884

Description: A restatement of noeta 32832 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 32867	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
2		ssltex1 32865	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
3		ssltss2 32868	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
4		ssltex2 32866	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
5		ssltsep 32869	. . 3 ⊢ (𝐴 <<s 𝐵 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)
6		noeta 32832	. . 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 1374	. 2 ⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
8		brsslt 32864	. . . . . 6 ⊢ (𝐴 <<s {𝑥} ↔ ((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧)))
9		df-3an 1082	. . . . . . 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 638	. . . . . 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 276	. . . . 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 5223	. . . . . . . . . 10 ⊢ {𝑥} ∈ V
14	12, 13	jctir 521	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → (𝐴 ∈ V ∧ {𝑥} ∈ V))
15	1	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → 𝐴 ⊆ No )
16		snssi 4648	. . . . . . . . . 10 ⊢ (𝑥 ∈ No → {𝑥} ⊆ No )
17	16	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → {𝑥} ⊆ No )
18	14, 15, 17	jca32 516	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No )))
19	18	biantrurd 533	. . . . . . 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 224	. . . . . 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 3440	. . . . . . . 8 ⊢ 𝑥 ∈ V
22		breq2 4966	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝑥))
23	21, 22	ralsn 4526	. . . . . . 7 ⊢ (∀𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ 𝑦 <s 𝑥)
24	23	ralbii 3132	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ ∀𝑦 ∈ 𝐴 𝑦 <s 𝑥)
25	20, 24	syl6bb 288	. . . . 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 284	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → (𝐴 <<s {𝑥} ↔ ∀𝑦 ∈ 𝐴 𝑦 <s 𝑥))
27		brsslt 32864	. . . . . . 7 ⊢ ({𝑥} <<s 𝐵 ↔ (({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)))
28		df-3an 1082	. . . . . . . 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 638	. . . . . . 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 276	. . . . . 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 520	. . . . . . . . 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 516	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → (({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No )))
35	34	biantrurd 533	. . . . . . 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 224	. . . . . 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 284	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ({𝑥} <<s 𝐵 ↔ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
38		breq1 4965	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 <s 𝑧 ↔ 𝑥 <s 𝑧))
39	38	ralbidv 3164	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝐵 𝑦 <s 𝑧 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧))
40	21, 39	ralsn 4526	. . . . 5 ⊢ (∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧)
41	37, 40	syl6bb 288	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ No ) → ({𝑥} <<s 𝐵 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧))
42	26, 41	3anbi12d 1429	. . 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 3259	. 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 258	1 ⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1080   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106  Vcvv 3437   ∪ cun 3857   ⊆ wss 3859  {csn 4472  ∪ cuni 4745   class class class wbr 4962   “ cima 5446  suc csuc 6068  ‘cfv 6225   No csur 32757   <s cslt 32758   bday cbday 32759   <<s csslt 32860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ord 6069  df-on 6070  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-1o 7953  df-2o 7954  df-no 32760  df-slt 32761  df-bday 32762  df-sslt 32861

This theorem is referenced by:  scutbdaybnd  32885