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Mirrors > Home > MPE Home > Th. List > Mathboxes > onnobdayg | Structured version Visualization version GIF version |
Description: Every ordinal maps to a surreal number of that birthday. (Contributed by RP, 21-Sep-2023.) |
Ref | Expression |
---|---|
onnobdayg | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onnog 43377 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No ) | |
2 | bdayval 27689 | . . 3 ⊢ ((𝐴 × {𝐵}) ∈ No → ( bday ‘(𝐴 × {𝐵})) = dom (𝐴 × {𝐵})) | |
3 | 1, 2 | syl 17 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = dom (𝐴 × {𝐵})) |
4 | simpr 484 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → 𝐵 ∈ {1o, 2o}) | |
5 | snnzg 4781 | . . 3 ⊢ (𝐵 ∈ {1o, 2o} → {𝐵} ≠ ∅) | |
6 | dmxp 5936 | . . 3 ⊢ ({𝐵} ≠ ∅ → dom (𝐴 × {𝐵}) = 𝐴) | |
7 | 4, 5, 6 | 3syl 18 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → dom (𝐴 × {𝐵}) = 𝐴) |
8 | 3, 7 | eqtrd 2773 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1535 ∈ wcel 2104 ≠ wne 2936 ∅c0 4339 {csn 4630 {cpr 4632 × cxp 5681 dom cdm 5683 Oncon0 6380 ‘cfv 6558 1oc1o 8492 2oc2o 8493 No csur 27680 bday cbday 27682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-fv 6566 df-no 27683 df-bday 27685 |
This theorem is referenced by: (None) |
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