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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 42782 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No ) | |
2 | bdayval 27568 | . . 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 4774 | . . 3 ⊢ (𝐵 ∈ {1o, 2o} → {𝐵} ≠ ∅) | |
6 | dmxp 5925 | . . 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 2767 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∅c0 4318 {csn 4624 {cpr 4626 × cxp 5670 dom cdm 5672 Oncon0 6363 ‘cfv 6542 1oc1o 8473 2oc2o 8474 No csur 27560 bday cbday 27562 |
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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-no 27563 df-bday 27565 |
This theorem is referenced by: (None) |
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