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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 3045 | . 2 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V) | |
2 | ssorduni 7798 | . . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
3 | ordeleqon 7801 | . . . . . . . 8 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On)) | |
4 | 2, 3 | sylib 218 | . . . . . . 7 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On)) |
5 | 4 | orcomd 871 | . . . . . 6 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = On ∨ ∪ 𝐴 ∈ On)) |
6 | 5 | ord 864 | . . . . 5 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → ∪ 𝐴 ∈ On)) |
7 | uniexr 7782 | . . . . 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 7797 | . . . 4 ⊢ ¬ On ∈ V | |
11 | uniexg 7759 | . . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V) | |
12 | eleq1 2827 | . . . . 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 847 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 Vcvv 3478 ⊆ wss 3963 ∪ cuni 4912 Ord word 6385 Oncon0 6386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: inaprc 10874 |
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