![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isnum2 | Structured version Visualization version GIF version |
Description: A way to express well-orderability without bound or distinct variables. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
isnum2 | ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardf2 9102 | . . . 4 ⊢ card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On | |
2 | 1 | fdmi 6301 | . . 3 ⊢ dom card = {𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦} |
3 | 2 | eleq2i 2851 | . 2 ⊢ (𝐴 ∈ dom card ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}) |
4 | relen 8246 | . . . . 5 ⊢ Rel ≈ | |
5 | 4 | brrelex2i 5407 | . . . 4 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ∈ V) |
6 | 5 | rexlimivw 3211 | . . 3 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → 𝐴 ∈ V) |
7 | breq2 4890 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝑥 ≈ 𝐴)) | |
8 | 7 | rexbidv 3237 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑥 ≈ 𝑦 ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)) |
9 | 6, 8 | elab3 3566 | . 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦} ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) |
10 | 3, 9 | bitri 267 | 1 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1601 ∈ wcel 2107 {cab 2763 ∃wrex 3091 Vcvv 3398 class class class wbr 4886 dom cdm 5355 Oncon0 5976 ≈ cen 8238 cardccrd 9094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-ord 5979 df-on 5980 df-fun 6137 df-fn 6138 df-f 6139 df-en 8242 df-card 9098 |
This theorem is referenced by: isnumi 9105 ennum 9106 xpnum 9110 cardval3 9111 dfac10c 9295 isfin7-2 9553 numth2 9628 inawinalem 9846 |
Copyright terms: Public domain | W3C validator |