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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 43420 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No ) | |
2 | bdayval 27683 | . . 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 4772 | . . 3 ⊢ (𝐵 ∈ {1o, 2o} → {𝐵} ≠ ∅) | |
6 | dmxp 5937 | . . 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 2776 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 ∅c0 4332 {csn 4624 {cpr 4626 × cxp 5681 dom cdm 5683 Oncon0 6382 ‘cfv 6559 1oc1o 8495 2oc2o 8496 No csur 27674 bday cbday 27676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 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-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-fv 6567 df-no 27677 df-bday 27679 |
This theorem is referenced by: (None) |
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