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Mirrors > Home > MPE Home > Th. List > ssonprc | Structured version Visualization version GIF version |
Description: Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.) |
Ref | Expression |
---|---|
ssonprc | ⊢ (𝐴 ⊆ On → (𝐴 ∉ V ↔ ∪ 𝐴 = On)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3092 | . 2 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V) | |
2 | ssorduni 7480 | . . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
3 | ordeleqon 7483 | . . . . . . . 8 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On)) | |
4 | 2, 3 | sylib 221 | . . . . . . 7 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On)) |
5 | 4 | orcomd 868 | . . . . . 6 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = On ∨ ∪ 𝐴 ∈ On)) |
6 | 5 | ord 861 | . . . . 5 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → ∪ 𝐴 ∈ On)) |
7 | uniexr 7465 | . . . . 5 ⊢ (∪ 𝐴 ∈ On → 𝐴 ∈ V) | |
8 | 6, 7 | syl6 35 | . . . 4 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → 𝐴 ∈ V)) |
9 | 8 | con1d 147 | . . 3 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V → ∪ 𝐴 = On)) |
10 | onprc 7479 | . . . 4 ⊢ ¬ On ∈ V | |
11 | uniexg 7446 | . . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V) | |
12 | eleq1 2877 | . . . . 5 ⊢ (∪ 𝐴 = On → (∪ 𝐴 ∈ V ↔ On ∈ V)) | |
13 | 11, 12 | syl5ib 247 | . . . 4 ⊢ (∪ 𝐴 = On → (𝐴 ∈ V → On ∈ V)) |
14 | 10, 13 | mtoi 202 | . . 3 ⊢ (∪ 𝐴 = On → ¬ 𝐴 ∈ V) |
15 | 9, 14 | impbid1 228 | . 2 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V ↔ ∪ 𝐴 = On)) |
16 | 1, 15 | syl5bb 286 | 1 ⊢ (𝐴 ⊆ On → (𝐴 ∉ V ↔ ∪ 𝐴 = On)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 Vcvv 3441 ⊆ wss 3881 ∪ cuni 4800 Ord word 6158 Oncon0 6159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 |
This theorem is referenced by: inaprc 10247 |
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