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| 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 3047 | . 2 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V) | |
| 2 | ssorduni 7799 | . . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
| 3 | ordeleqon 7802 | . . . . . . . 8 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On)) | |
| 4 | 2, 3 | sylib 218 | . . . . . . 7 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On)) | 
| 5 | 4 | orcomd 872 | . . . . . 6 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = On ∨ ∪ 𝐴 ∈ On)) | 
| 6 | 5 | ord 865 | . . . . 5 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → ∪ 𝐴 ∈ On)) | 
| 7 | uniexr 7783 | . . . . 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 7798 | . . . 4 ⊢ ¬ On ∈ V | |
| 11 | uniexg 7760 | . . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V) | |
| 12 | eleq1 2829 | . . . . 5 ⊢ (∪ 𝐴 = On → (∪ 𝐴 ∈ V ↔ On ∈ V)) | |
| 13 | 11, 12 | imbitrid 244 | . . . 4 ⊢ (∪ 𝐴 = On → (𝐴 ∈ V → On ∈ V)) | 
| 14 | 10, 13 | mtoi 199 | . . 3 ⊢ (∪ 𝐴 = On → ¬ 𝐴 ∈ V) | 
| 15 | 9, 14 | impbid1 225 | . 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 206 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 Vcvv 3480 ⊆ wss 3951 ∪ cuni 4907 Ord word 6383 Oncon0 6384 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 | 
| This theorem is referenced by: inaprc 10876 | 
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