Theorem etasslt 27876

Description: A restatement of noeta 27806 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 27851	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
2		ssltex1 27849	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
3	1, 2	jca 511	. . . . 5 ⊢ (𝐴 <<s 𝐵 → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
4		ssltss2 27852	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
5		ssltex2 27850	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
6	4, 5	jca 511	. . . . 5 ⊢ (𝐴 <<s 𝐵 → (𝐵 ⊆ No ∧ 𝐵 ∈ V))
7		ssltsep 27853	. . . . 5 ⊢ (𝐴 <<s 𝐵 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)
8	3, 6, 7	3jca 1128	. . . 4 ⊢ (𝐴 <<s 𝐵 → ((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
9	8	3ad2ant1 1133	. . 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 1150	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂))
11		noeta 27806	. . 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 583	. 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 725	. . . . . . . 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 5449	. . . . . . . 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 725	. . . . . . . 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 4833	. . . . . . . . 9 ⊢ (𝑥 ∈ No → {𝑥} ⊆ No )
18	17	ad2antrl 727	. . . . . . . 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 1221	. . . . . . . . 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 3492	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
21		breq2 5170	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝑥))
22	20, 21	ralsn 4705	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ 𝑦 <s 𝑥)
23	22	ralbii 3099	. . . . . . . . 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 1128	. . . . . . 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 27848	. . . . . . 7 ⊢ (𝐴 <<s {𝑥} ↔ ((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑥} ⊆ No ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧)))
27	15, 25, 26	sylanbrc 582	. . . . . 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 725	. . . . . . . 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 725	. . . . . . . 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 1222	. . . . . . . . 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 5169	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 <s 𝑧 ↔ 𝑥 <s 𝑧))
33	32	ralbidv 3184	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝐵 𝑦 <s 𝑧 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧))
34	20, 33	ralsn 4705	. . . . . . . . 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 1128	. . . . . . 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 27848	. . . . . . 7 ⊢ ({𝑥} <<s 𝐵 ↔ (({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)))
38	29, 36, 37	sylanbrc 582	. . . . . 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 1223	. . . . . 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 1128	. . . . 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 3174	. . 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 1132	. 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 1087   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976  {csn 4648   class class class wbr 5166   “ cima 5703  Oncon0 6395  ‘cfv 6573   No csur 27702   <s cslt 27703   bday cbday 27704   <<s csslt 27843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844

This theorem is referenced by:  etasslt2  27877  scutbdaybnd  27878