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Mirrors > Home > MPE Home > Th. List > Mathboxes > madebdaylemold | Structured version Visualization version GIF version |
Description: Lemma for madebday 34080. If the inductive hypothesis of madebday 34080 is satisfied, the converse of oldbdayim 34071 holds. (Contributed by Scott Fenton, 19-Aug-2024.) |
Ref | Expression |
---|---|
madebdaylemold | ⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6774 | . . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ( bday ‘𝑦) = ( bday ‘𝑋)) | |
2 | 1 | sseq1d 3952 | . . . . . . . 8 ⊢ (𝑦 = 𝑋 → (( bday ‘𝑦) ⊆ 𝑏 ↔ ( bday ‘𝑋) ⊆ 𝑏)) |
3 | eleq1 2826 | . . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦 ∈ ( M ‘𝑏) ↔ 𝑋 ∈ ( M ‘𝑏))) | |
4 | 2, 3 | imbi12d 345 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → ((( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ↔ (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏)))) |
5 | 4 | rspcv 3557 | . . . . . 6 ⊢ (𝑋 ∈ No → (∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) → (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏)))) |
6 | 5 | ralimdv 3109 | . . . . 5 ⊢ (𝑋 ∈ No → (∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) → ∀𝑏 ∈ 𝐴 (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏)))) |
7 | 6 | impcom 408 | . . . 4 ⊢ ((∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → ∀𝑏 ∈ 𝐴 (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏))) |
8 | rexim 3172 | . . . 4 ⊢ (∀𝑏 ∈ 𝐴 (( bday ‘𝑋) ⊆ 𝑏 → 𝑋 ∈ ( M ‘𝑏)) → (∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏 → ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏))) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ ((∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏 → ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏))) |
10 | 9 | 3adant1 1129 | . 2 ⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏 → ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏))) |
11 | bdayelon 33971 | . . . 4 ⊢ ( bday ‘𝑋) ∈ On | |
12 | onelssex 33661 | . . . 4 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝐴 ∈ On) → (( bday ‘𝑋) ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏)) | |
13 | 11, 12 | mpan 687 | . . 3 ⊢ (𝐴 ∈ On → (( bday ‘𝑋) ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏)) |
14 | 13 | 3ad2ant1 1132 | . 2 ⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐴 ( bday ‘𝑋) ⊆ 𝑏)) |
15 | elold 34053 | . . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏))) | |
16 | 15 | 3ad2ant1 1132 | . 2 ⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏))) |
17 | 10, 14, 16 | 3imtr4d 294 | 1 ⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 Oncon0 6266 ‘cfv 6433 No csur 33843 bday cbday 33845 M cmade 34026 O cold 34027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-1o 8297 df-2o 8298 df-no 33846 df-slt 33847 df-bday 33848 df-sslt 33976 df-scut 33978 df-made 34031 df-old 34032 |
This theorem is referenced by: madebdaylemlrcut 34079 oldbday 34081 |
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