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Mirrors > Home > MPE Home > Th. List > fin4i | Structured version Visualization version GIF version |
Description: Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.) |
Ref | Expression |
---|---|
fin4i | ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ¬ 𝐴 ∈ FinIV) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfin4 10289 | . . 3 ⊢ (𝐴 ∈ FinIV → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))) | |
2 | 1 | ibi 267 | . 2 ⊢ (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
3 | relen 8941 | . . . . 5 ⊢ Rel ≈ | |
4 | 3 | brrelex1i 5723 | . . . 4 ⊢ (𝑋 ≈ 𝐴 → 𝑋 ∈ V) |
5 | 4 | adantl 481 | . . 3 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → 𝑋 ∈ V) |
6 | psseq1 4080 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ⊊ 𝐴 ↔ 𝑋 ⊊ 𝐴)) | |
7 | breq1 5142 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≈ 𝐴 ↔ 𝑋 ≈ 𝐴)) | |
8 | 6, 7 | anbi12d 630 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) ↔ (𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴))) |
9 | 8 | spcegv 3579 | . . 3 ⊢ (𝑋 ∈ V → ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))) |
10 | 5, 9 | mpcom 38 | . 2 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
11 | 2, 10 | nsyl3 138 | 1 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ¬ 𝐴 ∈ FinIV) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3466 ⊊ wpss 3942 class class class wbr 5139 ≈ cen 8933 FinIVcfin4 10272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-xp 5673 df-rel 5674 df-en 8937 df-fin4 10279 |
This theorem is referenced by: fin4en1 10301 ssfin4 10302 ominf4 10304 isfin4p1 10307 |
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