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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 27888 | . . 3 ⊢ ( bday ‘ 0s ) = ∅ | |
2 | 0elold.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0s ) | |
3 | 2 | neneqd 2943 | . . . . 5 ⊢ (𝜑 → ¬ 𝐴 = 0s ) |
4 | 0elold.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No ) | |
5 | bday0b 27890 | . . . . . 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 27836 | . . . . 5 ⊢ ( bday ‘𝐴) ∈ On | |
9 | on0eqel 6510 | . . . . 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 2843 | . 2 ⊢ (𝜑 → ( bday ‘ 0s ) ∈ ( bday ‘𝐴)) |
14 | 0sno 27886 | . . 3 ⊢ 0s ∈ No | |
15 | oldbday 27954 | . . 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 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 Oncon0 6386 ‘cfv 6563 No csur 27699 bday cbday 27701 0s c0s 27882 O cold 27897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-1o 8505 df-2o 8506 df-no 27702 df-slt 27703 df-bday 27704 df-sslt 27841 df-scut 27843 df-0s 27884 df-made 27901 df-old 27902 df-left 27904 df-right 27905 |
This theorem is referenced by: 0elleft 27963 0elright 27964 |
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