Theorem etasslt 27873

Description: A restatement of noeta 27803 using set less-than. (Contributed by Scott Fenton, 10-Aug-2024.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑂

Proof of Theorem etasslt

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

Step	Hyp	Ref	Expression
1		ssltss1 27848	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
2		ssltex1 27846	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
3	1, 2	jca 511	. . . . 5 ⊢ (𝐴 <<s 𝐵 → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
4		ssltss2 27849	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
5		ssltex2 27847	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
6	4, 5	jca 511	. . . . 5 ⊢ (𝐴 <<s 𝐵 → (𝐵 ⊆ No ∧ 𝐵 ∈ V))
7		ssltsep 27850	. . . . 5 ⊢ (𝐴 <<s 𝐵 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)
8	3, 6, 7	3jca 1127	. . . 4 ⊢ (𝐴 <<s 𝐵 → ((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
9	8	3ad2ant1 1132	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → ((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
10		3simpc 1149	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂))
11		noeta 27803	. . 3 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝑦 <s 𝑧) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ∃𝑥 ∈ No (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))
12	9, 10, 11	syl2anc 584	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → ∃𝑥 ∈ No (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))
13	2	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → 𝐴 ∈ V)
14		vsnex 5440	. . . . . . . 8 ⊢ {𝑥} ∈ V
15	13, 14	jctir 520	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → (𝐴 ∈ V ∧ {𝑥} ∈ V))
16	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → 𝐴 ⊆ No )
17		snssi 4813	. . . . . . . . 9 ⊢ (𝑥 ∈ No → {𝑥} ⊆ No )
18	17	ad2antrl 728	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → {𝑥} ⊆ No )
19		simprr1 1220	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ∀𝑦 ∈ 𝐴 𝑦 <s 𝑥)
20		vex 3482	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
21		breq2 5152	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝑥))
22	20, 21	ralsn 4686	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ 𝑦 <s 𝑥)
23	22	ralbii 3091	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ ∀𝑦 ∈ 𝐴 𝑦 <s 𝑥)
24	19, 23	sylibr 234	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧)
25	16, 18, 24	3jca 1127	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → (𝐴 ⊆ No ∧ {𝑥} ⊆ No ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧))
26		brsslt 27845	. . . . . . 7 ⊢ (𝐴 <<s {𝑥} ↔ ((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧)))
27	15, 25, 26	sylanbrc 583	. . . . . 6 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → 𝐴 <<s {𝑥})
28	5	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → 𝐵 ∈ V)
29	28, 14	jctil 519	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ({𝑥} ∈ V ∧ 𝐵 ∈ V))
30	4	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → 𝐵 ⊆ No )
31		simprr2 1221	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧)
32		breq1 5151	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 <s 𝑧 ↔ 𝑥 <s 𝑧))
33	32	ralbidv 3176	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝐵 𝑦 <s 𝑧 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧))
34	20, 33	ralsn 4686	. . . . . . . . 9 ⊢ (∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧)
35	31, 34	sylibr 234	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)
36	18, 30, 35	3jca 1127	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ({𝑥} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
37		brsslt 27845	. . . . . . 7 ⊢ ({𝑥} <<s 𝐵 ↔ (({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)))
38	29, 36, 37	sylanbrc 583	. . . . . 6 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → {𝑥} <<s 𝐵)
39		simprr3 1222	. . . . . 6 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → ( bday ‘𝑥) ⊆ 𝑂)
40	27, 38, 39	3jca 1127	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))
41	40	expr 456	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ 𝑥 ∈ No ) → ((∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂) → (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂)))
42	41	reximdva 3166	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) → (∃𝑥 ∈ No (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂) → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂)))
43	42	3adant3 1131	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → (∃𝑥 ∈ No (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂) → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂)))
44	12, 43	mpd 15	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ∪ cun 3961   ⊆ wss 3963  {csn 4631   class class class wbr 5148   “ cima 5692  Oncon0 6386  ‘cfv 6563   No csur 27699   <s cslt 27700   bday cbday 27701   <<s csslt 27840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841

This theorem is referenced by:  etasslt2  27874  scutbdaybnd  27875