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Theorem etasslt 32247
 Description: A restatement of noeta 32195 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 “ (𝐴𝐵))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem etasslt
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltss1 32230 . . 3 (𝐴 <<s 𝐵𝐴 No )
2 ssltex1 32228 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
3 ssltss2 32231 . . 3 (𝐴 <<s 𝐵𝐵 No )
4 ssltex2 32229 . . 3 (𝐴 <<s 𝐵𝐵 ∈ V)
5 ssltsep 32232 . . 3 (𝐴 <<s 𝐵 → ∀𝑦𝐴𝑧𝐵 𝑦 <s 𝑧)
6 noeta 32195 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑦𝐴𝑧𝐵 𝑦 <s 𝑧) → ∃𝑥 No (∀𝑦𝐴 𝑦 <s 𝑥 ∧ ∀𝑧𝐵 𝑥 <s 𝑧 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))
71, 2, 3, 4, 5, 6syl221anc 1488 . 2 (𝐴 <<s 𝐵 → ∃𝑥 No (∀𝑦𝐴 𝑦 <s 𝑥 ∧ ∀𝑧𝐵 𝑥 <s 𝑧 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))
8 brsslt 32227 . . . . . 6 (𝐴 <<s {𝑥} ↔ ((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 No ∧ {𝑥} ⊆ No ∧ ∀𝑦𝐴𝑧 ∈ {𝑥}𝑦 <s 𝑧)))
9 df-3an 1074 . . . . . . 7 ((𝐴 No ∧ {𝑥} ⊆ No ∧ ∀𝑦𝐴𝑧 ∈ {𝑥}𝑦 <s 𝑧) ↔ ((𝐴 No ∧ {𝑥} ⊆ No ) ∧ ∀𝑦𝐴𝑧 ∈ {𝑥}𝑦 <s 𝑧))
109bianass 877 . . . . . 6 (((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 No ∧ {𝑥} ⊆ No ∧ ∀𝑦𝐴𝑧 ∈ {𝑥}𝑦 <s 𝑧)) ↔ (((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 No ∧ {𝑥} ⊆ No )) ∧ ∀𝑦𝐴𝑧 ∈ {𝑥}𝑦 <s 𝑧))
118, 10bitri 264 . . . . 5 (𝐴 <<s {𝑥} ↔ (((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 No ∧ {𝑥} ⊆ No )) ∧ ∀𝑦𝐴𝑧 ∈ {𝑥}𝑦 <s 𝑧))
122adantr 472 . . . . . . . . . 10 ((𝐴 <<s 𝐵𝑥 No ) → 𝐴 ∈ V)
13 snex 5057 . . . . . . . . . 10 {𝑥} ∈ V
1412, 13jctir 562 . . . . . . . . 9 ((𝐴 <<s 𝐵𝑥 No ) → (𝐴 ∈ V ∧ {𝑥} ∈ V))
151adantr 472 . . . . . . . . 9 ((𝐴 <<s 𝐵𝑥 No ) → 𝐴 No )
16 snssi 4484 . . . . . . . . . 10 (𝑥 No → {𝑥} ⊆ No )
1716adantl 473 . . . . . . . . 9 ((𝐴 <<s 𝐵𝑥 No ) → {𝑥} ⊆ No )
1814, 15, 17jca32 559 . . . . . . . 8 ((𝐴 <<s 𝐵𝑥 No ) → ((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 No ∧ {𝑥} ⊆ No )))
1918biantrurd 530 . . . . . . 7 ((𝐴 <<s 𝐵𝑥 No ) → (∀𝑦𝐴𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ (((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 No ∧ {𝑥} ⊆ No )) ∧ ∀𝑦𝐴𝑧 ∈ {𝑥}𝑦 <s 𝑧)))
2019bicomd 213 . . . . . 6 ((𝐴 <<s 𝐵𝑥 No ) → ((((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 No ∧ {𝑥} ⊆ No )) ∧ ∀𝑦𝐴𝑧 ∈ {𝑥}𝑦 <s 𝑧) ↔ ∀𝑦𝐴𝑧 ∈ {𝑥}𝑦 <s 𝑧))
21 vex 3343 . . . . . . . 8 𝑥 ∈ V
22 breq2 4808 . . . . . . . 8 (𝑧 = 𝑥 → (𝑦 <s 𝑧𝑦 <s 𝑥))
2321, 22ralsn 4366 . . . . . . 7 (∀𝑧 ∈ {𝑥}𝑦 <s 𝑧𝑦 <s 𝑥)
2423ralbii 3118 . . . . . 6 (∀𝑦𝐴𝑧 ∈ {𝑥}𝑦 <s 𝑧 ↔ ∀𝑦𝐴 𝑦 <s 𝑥)
2520, 24syl6bb 276 . . . . 5 ((𝐴 <<s 𝐵𝑥 No ) → ((((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 No ∧ {𝑥} ⊆ No )) ∧ ∀𝑦𝐴𝑧 ∈ {𝑥}𝑦 <s 𝑧) ↔ ∀𝑦𝐴 𝑦 <s 𝑥))
2611, 25syl5bb 272 . . . 4 ((𝐴 <<s 𝐵𝑥 No ) → (𝐴 <<s {𝑥} ↔ ∀𝑦𝐴 𝑦 <s 𝑥))
27 brsslt 32227 . . . . . . 7 ({𝑥} <<s 𝐵 ↔ (({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No 𝐵 No ∧ ∀𝑦 ∈ {𝑥}∀𝑧𝐵 𝑦 <s 𝑧)))
28 df-3an 1074 . . . . . . . 8 (({𝑥} ⊆ No 𝐵 No ∧ ∀𝑦 ∈ {𝑥}∀𝑧𝐵 𝑦 <s 𝑧) ↔ (({𝑥} ⊆ No 𝐵 No ) ∧ ∀𝑦 ∈ {𝑥}∀𝑧𝐵 𝑦 <s 𝑧))
2928bianass 877 . . . . . . 7 ((({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No 𝐵 No ∧ ∀𝑦 ∈ {𝑥}∀𝑧𝐵 𝑦 <s 𝑧)) ↔ ((({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No 𝐵 No )) ∧ ∀𝑦 ∈ {𝑥}∀𝑧𝐵 𝑦 <s 𝑧))
3027, 29bitri 264 . . . . . 6 ({𝑥} <<s 𝐵 ↔ ((({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No 𝐵 No )) ∧ ∀𝑦 ∈ {𝑥}∀𝑧𝐵 𝑦 <s 𝑧))
314adantr 472 . . . . . . . . . 10 ((𝐴 <<s 𝐵𝑥 No ) → 𝐵 ∈ V)
3231, 13jctil 561 . . . . . . . . 9 ((𝐴 <<s 𝐵𝑥 No ) → ({𝑥} ∈ V ∧ 𝐵 ∈ V))
333adantr 472 . . . . . . . . 9 ((𝐴 <<s 𝐵𝑥 No ) → 𝐵 No )
3432, 17, 33jca32 559 . . . . . . . 8 ((𝐴 <<s 𝐵𝑥 No ) → (({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No 𝐵 No )))
3534biantrurd 530 . . . . . . 7 ((𝐴 <<s 𝐵𝑥 No ) → (∀𝑦 ∈ {𝑥}∀𝑧𝐵 𝑦 <s 𝑧 ↔ ((({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No 𝐵 No )) ∧ ∀𝑦 ∈ {𝑥}∀𝑧𝐵 𝑦 <s 𝑧)))
3635bicomd 213 . . . . . 6 ((𝐴 <<s 𝐵𝑥 No ) → (((({𝑥} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑥} ⊆ No 𝐵 No )) ∧ ∀𝑦 ∈ {𝑥}∀𝑧𝐵 𝑦 <s 𝑧) ↔ ∀𝑦 ∈ {𝑥}∀𝑧𝐵 𝑦 <s 𝑧))
3730, 36syl5bb 272 . . . . 5 ((𝐴 <<s 𝐵𝑥 No ) → ({𝑥} <<s 𝐵 ↔ ∀𝑦 ∈ {𝑥}∀𝑧𝐵 𝑦 <s 𝑧))
38 breq1 4807 . . . . . . 7 (𝑦 = 𝑥 → (𝑦 <s 𝑧𝑥 <s 𝑧))
3938ralbidv 3124 . . . . . 6 (𝑦 = 𝑥 → (∀𝑧𝐵 𝑦 <s 𝑧 ↔ ∀𝑧𝐵 𝑥 <s 𝑧))
4021, 39ralsn 4366 . . . . 5 (∀𝑦 ∈ {𝑥}∀𝑧𝐵 𝑦 <s 𝑧 ↔ ∀𝑧𝐵 𝑥 <s 𝑧)
4137, 40syl6bb 276 . . . 4 ((𝐴 <<s 𝐵𝑥 No ) → ({𝑥} <<s 𝐵 ↔ ∀𝑧𝐵 𝑥 <s 𝑧))
4226, 413anbi12d 1549 . . 3 ((𝐴 <<s 𝐵𝑥 No ) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))) ↔ (∀𝑦𝐴 𝑦 <s 𝑥 ∧ ∀𝑧𝐵 𝑥 <s 𝑧 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵)))))
4342rexbidva 3187 . 2 (𝐴 <<s 𝐵 → (∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))) ↔ ∃𝑥 No (∀𝑦𝐴 𝑦 <s 𝑥 ∧ ∀𝑧𝐵 𝑥 <s 𝑧 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵)))))
447, 43mpbird 247 1 (𝐴 <<s 𝐵 → ∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051  Vcvv 3340   ∪ cun 3713   ⊆ wss 3715  {csn 4321  ∪ cuni 4588   class class class wbr 4804   “ cima 5269  suc csuc 5886  ‘cfv 6049   No csur 32120
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