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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 3124 | . 2 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V) | |
2 | ssorduni 7500 | . . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
3 | ordeleqon 7503 | . . . . . . . 8 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On)) | |
4 | 2, 3 | sylib 220 | . . . . . . 7 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On)) |
5 | 4 | orcomd 867 | . . . . . 6 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = On ∨ ∪ 𝐴 ∈ On)) |
6 | 5 | ord 860 | . . . . 5 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → ∪ 𝐴 ∈ On)) |
7 | uniexr 7485 | . . . . 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 7499 | . . . 4 ⊢ ¬ On ∈ V | |
11 | uniexg 7466 | . . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V) | |
12 | eleq1 2900 | . . . . 5 ⊢ (∪ 𝐴 = On → (∪ 𝐴 ∈ V ↔ On ∈ V)) | |
13 | 11, 12 | syl5ib 246 | . . . 4 ⊢ (∪ 𝐴 = On → (𝐴 ∈ V → On ∈ V)) |
14 | 10, 13 | mtoi 201 | . . 3 ⊢ (∪ 𝐴 = On → ¬ 𝐴 ∈ V) |
15 | 9, 14 | impbid1 227 | . 2 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V ↔ ∪ 𝐴 = On)) |
16 | 1, 15 | syl5bb 285 | 1 ⊢ (𝐴 ⊆ On → (𝐴 ∉ V ↔ ∪ 𝐴 = On)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 Vcvv 3494 ⊆ wss 3936 ∪ cuni 4838 Ord word 6190 Oncon0 6191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 |
This theorem is referenced by: inaprc 10258 |
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