Theorem etasslt 33934

Description: A restatement of noeta 33873 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 33910	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
2		ssltex1 33908	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
3	1, 2	jca 511	. . . . 5 ⊢ (𝐴 <<s 𝐵 → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
4		ssltss2 33911	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
5		ssltex2 33909	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
6	4, 5	jca 511	. . . . 5 ⊢ (𝐴 <<s 𝐵 → (𝐵 ⊆ No ∧ 𝐵 ∈ V))
7		ssltsep 33912	. . . . 5 ⊢ (𝐴 <<s 𝐵 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝑦 <s 𝑧)
8	3, 6, 7	3jca 1126	. . . 4 ⊢ (𝐴 <<s 𝐵 → ((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝑦 <s 𝑧))
9	8	3ad2ant1 1131	. . 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 1148	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂))
11		noeta 33873	. . 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 722	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On) ∧ (𝑥 ∈ No ∧ (∀𝑦 ∈ 𝐴 𝑦 <s 𝑥 ∧ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧 ∧ ( bday ‘𝑥) ⊆ 𝑂))) → 𝐴 ∈ V)
14		snex 5349	. . . . . . . 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 722	. . . . . . . 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 4738	. . . . . . . . 9 ⊢ (𝑥 ∈ No → {𝑥} ⊆ No )
18	17	ad2antrl 724	. . . . . . . 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 1219	. . . . . . . . 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 3426	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
21		breq2 5074	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝑥))
22	20, 21	ralsn 4614	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ 𝑦 <s 𝑥)
23	22	ralbii 3090	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ ∀𝑦 ∈ 𝐴 𝑦 <s 𝑥)
24	19, 23	sylibr 233	. . . . . . . 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 1126	. . . . . . 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 33907	. . . . . . 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 722	. . . . . . . 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 722	. . . . . . . 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 1220	. . . . . . . . 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 5073	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 <s 𝑧 ↔ 𝑥 <s 𝑧))
33	32	ralbidv 3120	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝐵 𝑦 <s 𝑧 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧))
34	20, 33	ralsn 4614	. . . . . . . . 9 ⊢ (∀𝑦 ∈ {𝑥}∀𝑧 ∈ 𝐵 𝑦 <s 𝑧 ↔ ∀𝑧 ∈ 𝐵 𝑥 <s 𝑧)
35	31, 34	sylibr 233	. . . . . . . 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 1126	. . . . . . 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 33907	. . . . . . 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 1221	. . . . . 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 1126	. . . . 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 3202	. . 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 1130	. 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 1085   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  {csn 4558   class class class wbr 5070   “ cima 5583  Oncon0 6251  ‘cfv 6418   No csur 33770   <s cslt 33771   bday cbday 33772   <<s csslt 33902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775  df-sslt 33903

This theorem is referenced by:  etasslt2  33935  scutbdaybnd  33936