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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 9565 | . . 3 ⊢ (𝐴 ∈ FinIV → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))) | |
2 | 1 | ibi 268 | . 2 ⊢ (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
3 | relen 8362 | . . . . 5 ⊢ Rel ≈ | |
4 | 3 | brrelex1i 5494 | . . . 4 ⊢ (𝑋 ≈ 𝐴 → 𝑋 ∈ V) |
5 | 4 | adantl 482 | . . 3 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → 𝑋 ∈ V) |
6 | psseq1 3985 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ⊊ 𝐴 ↔ 𝑋 ⊊ 𝐴)) | |
7 | breq1 4965 | . . . . 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 3540 | . . 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 140 | 1 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ¬ 𝐴 ∈ FinIV) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1522 ∃wex 1761 ∈ wcel 2081 Vcvv 3437 ⊊ wpss 3860 class class class wbr 4962 ≈ cen 8354 FinIVcfin4 9548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-br 4963 df-opab 5025 df-xp 5449 df-rel 5450 df-en 8358 df-fin4 9555 |
This theorem is referenced by: fin4en1 9577 ssfin4 9578 ominf4 9580 isfin4p1 9583 |
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