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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 3051 | . 2 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V) | |
2 | ssorduni 7718 | . . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
3 | ordeleqon 7721 | . . . . . . . 8 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On)) | |
4 | 2, 3 | sylib 217 | . . . . . . 7 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On)) |
5 | 4 | orcomd 870 | . . . . . 6 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = On ∨ ∪ 𝐴 ∈ On)) |
6 | 5 | ord 863 | . . . . 5 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → ∪ 𝐴 ∈ On)) |
7 | uniexr 7702 | . . . . 5 ⊢ (∪ 𝐴 ∈ On → 𝐴 ∈ V) | |
8 | 6, 7 | syl6 35 | . . . 4 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → 𝐴 ∈ V)) |
9 | 8 | con1d 145 | . . 3 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V → ∪ 𝐴 = On)) |
10 | onprc 7717 | . . . 4 ⊢ ¬ On ∈ V | |
11 | uniexg 7682 | . . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V) | |
12 | eleq1 2826 | . . . . 5 ⊢ (∪ 𝐴 = On → (∪ 𝐴 ∈ V ↔ On ∈ V)) | |
13 | 11, 12 | imbitrid 243 | . . . 4 ⊢ (∪ 𝐴 = On → (𝐴 ∈ V → On ∈ V)) |
14 | 10, 13 | mtoi 198 | . . 3 ⊢ (∪ 𝐴 = On → ¬ 𝐴 ∈ V) |
15 | 9, 14 | impbid1 224 | . 2 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V ↔ ∪ 𝐴 = On)) |
16 | 1, 15 | bitrid 283 | 1 ⊢ (𝐴 ⊆ On → (𝐴 ∉ V ↔ ∪ 𝐴 = On)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∉ wnel 3050 Vcvv 3448 ⊆ wss 3915 ∪ cuni 4870 Ord word 6321 Oncon0 6322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-ord 6325 df-on 6326 |
This theorem is referenced by: inaprc 10779 |
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