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| Mirrors > Home > MPE Home > Th. List > 0elold | Structured version Visualization version GIF version | ||
| Description: Zero is in the old set of any non-zero number. (Contributed by Scott Fenton, 13-Mar-2025.) | 
| Ref | Expression | 
|---|---|
| 0elold.1 | ⊢ (𝜑 → 𝐴 ∈ No ) | 
| 0elold.2 | ⊢ (𝜑 → 𝐴 ≠ 0s ) | 
| Ref | Expression | 
|---|---|
| 0elold | ⊢ (𝜑 → 0s ∈ ( O ‘( bday ‘𝐴))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bday0s 27874 | . . 3 ⊢ ( bday ‘ 0s ) = ∅ | |
| 2 | 0elold.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0s ) | |
| 3 | 2 | neneqd 2944 | . . . . 5 ⊢ (𝜑 → ¬ 𝐴 = 0s ) | 
| 4 | 0elold.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No ) | |
| 5 | bday0b 27876 | . . . . . 6 ⊢ (𝐴 ∈ No → (( bday ‘𝐴) = ∅ ↔ 𝐴 = 0s )) | |
| 6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → (( bday ‘𝐴) = ∅ ↔ 𝐴 = 0s )) | 
| 7 | 3, 6 | mtbird 325 | . . . 4 ⊢ (𝜑 → ¬ ( bday ‘𝐴) = ∅) | 
| 8 | bdayelon 27822 | . . . . 5 ⊢ ( bday ‘𝐴) ∈ On | |
| 9 | on0eqel 6507 | . . . . 5 ⊢ (( bday ‘𝐴) ∈ On → (( bday ‘𝐴) = ∅ ∨ ∅ ∈ ( bday ‘𝐴))) | |
| 10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ (( bday ‘𝐴) = ∅ ∨ ∅ ∈ ( bday ‘𝐴)) | 
| 11 | orel1 888 | . . . 4 ⊢ (¬ ( bday ‘𝐴) = ∅ → ((( bday ‘𝐴) = ∅ ∨ ∅ ∈ ( bday ‘𝐴)) → ∅ ∈ ( bday ‘𝐴))) | |
| 12 | 7, 10, 11 | mpisyl 21 | . . 3 ⊢ (𝜑 → ∅ ∈ ( bday ‘𝐴)) | 
| 13 | 1, 12 | eqeltrid 2844 | . 2 ⊢ (𝜑 → ( bday ‘ 0s ) ∈ ( bday ‘𝐴)) | 
| 14 | 0sno 27872 | . . 3 ⊢ 0s ∈ No | |
| 15 | oldbday 27940 | . . 3 ⊢ ((( bday ‘𝐴) ∈ On ∧ 0s ∈ No ) → ( 0s ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘ 0s ) ∈ ( bday ‘𝐴))) | |
| 16 | 8, 14, 15 | mp2an 692 | . 2 ⊢ ( 0s ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘ 0s ) ∈ ( bday ‘𝐴)) | 
| 17 | 13, 16 | sylibr 234 | 1 ⊢ (𝜑 → 0s ∈ ( O ‘( bday ‘𝐴))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 Oncon0 6383 ‘cfv 6560 No csur 27685 bday cbday 27687 0s c0s 27868 O cold 27883 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-1o 8507 df-2o 8508 df-no 27688 df-slt 27689 df-bday 27690 df-sslt 27827 df-scut 27829 df-0s 27870 df-made 27887 df-old 27888 df-left 27890 df-right 27891 | 
| This theorem is referenced by: 0elleft 27949 0elright 27950 | 
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