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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 9713 | . . 3 ⊢ (𝐴 ∈ FinIV → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))) | |
2 | 1 | ibi 269 | . 2 ⊢ (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
3 | relen 8508 | . . . . 5 ⊢ Rel ≈ | |
4 | 3 | brrelex1i 5602 | . . . 4 ⊢ (𝑋 ≈ 𝐴 → 𝑋 ∈ V) |
5 | 4 | adantl 484 | . . 3 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → 𝑋 ∈ V) |
6 | psseq1 4063 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ⊊ 𝐴 ↔ 𝑋 ⊊ 𝐴)) | |
7 | breq1 5061 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≈ 𝐴 ↔ 𝑋 ≈ 𝐴)) | |
8 | 6, 7 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) ↔ (𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴))) |
9 | 8 | spcegv 3596 | . . 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 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 ⊊ wpss 3936 class class class wbr 5058 ≈ cen 8500 FinIVcfin4 9696 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-en 8504 df-fin4 9703 |
This theorem is referenced by: fin4en1 9725 ssfin4 9726 ominf4 9728 isfin4p1 9731 |
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