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Mirrors > Home > MPE Home > Th. List > snfiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of snfi 9081 as of 13-Jan-2025. (Contributed by NM, 4-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
snfiOLD | ⊢ {𝐴} ∈ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 8676 | . . . 4 ⊢ 1o ∈ ω | |
2 | ensn1g 9060 | . . . 4 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o) | |
3 | breq2 5151 | . . . . 5 ⊢ (𝑥 = 1o → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ 1o)) | |
4 | 3 | rspcev 3621 | . . . 4 ⊢ ((1o ∈ ω ∧ {𝐴} ≈ 1o) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
5 | 1, 2, 4 | sylancr 587 | . . 3 ⊢ (𝐴 ∈ V → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
6 | snprc 4721 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
7 | en0 9056 | . . . . 5 ⊢ ({𝐴} ≈ ∅ ↔ {𝐴} = ∅) | |
8 | peano1 7910 | . . . . . 6 ⊢ ∅ ∈ ω | |
9 | breq2 5151 | . . . . . . 7 ⊢ (𝑥 = ∅ → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ ∅)) | |
10 | 9 | rspcev 3621 | . . . . . 6 ⊢ ((∅ ∈ ω ∧ {𝐴} ≈ ∅) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
11 | 8, 10 | mpan 690 | . . . . 5 ⊢ ({𝐴} ≈ ∅ → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
12 | 7, 11 | sylbir 235 | . . . 4 ⊢ ({𝐴} = ∅ → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
13 | 6, 12 | sylbi 217 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
14 | 5, 13 | pm2.61i 182 | . 2 ⊢ ∃𝑥 ∈ ω {𝐴} ≈ 𝑥 |
15 | isfi 9014 | . 2 ⊢ ({𝐴} ∈ Fin ↔ ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) | |
16 | 14, 15 | mpbir 231 | 1 ⊢ {𝐴} ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 Vcvv 3477 ∅c0 4338 {csn 4630 class class class wbr 5147 ωcom 7886 1oc1o 8497 ≈ cen 8980 Fincfn 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-mo 2537 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-om 7887 df-1o 8504 df-en 8984 df-fin 8987 |
This theorem is referenced by: (None) |
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