![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > enp1i | Structured version Visualization version GIF version |
Description: Proof induction for en2i 8146 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Ref | Expression |
---|---|
enp1i.1 | ⊢ 𝑀 ∈ ω |
enp1i.2 | ⊢ 𝑁 = suc 𝑀 |
enp1i.3 | ⊢ ((𝐴 ∖ {𝑥}) ≈ 𝑀 → 𝜑) |
enp1i.4 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
enp1i | ⊢ (𝐴 ≈ 𝑁 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsuceq0 5948 | . . . . 5 ⊢ suc 𝑀 ≠ ∅ | |
2 | breq1 4787 | . . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ≈ 𝑁 ↔ ∅ ≈ 𝑁)) | |
3 | enp1i.2 | . . . . . . . 8 ⊢ 𝑁 = suc 𝑀 | |
4 | ensym 8157 | . . . . . . . . 9 ⊢ (∅ ≈ 𝑁 → 𝑁 ≈ ∅) | |
5 | en0 8171 | . . . . . . . . 9 ⊢ (𝑁 ≈ ∅ ↔ 𝑁 = ∅) | |
6 | 4, 5 | sylib 208 | . . . . . . . 8 ⊢ (∅ ≈ 𝑁 → 𝑁 = ∅) |
7 | 3, 6 | syl5eqr 2818 | . . . . . . 7 ⊢ (∅ ≈ 𝑁 → suc 𝑀 = ∅) |
8 | 2, 7 | syl6bi 243 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ≈ 𝑁 → suc 𝑀 = ∅)) |
9 | 8 | necon3ad 2955 | . . . . 5 ⊢ (𝐴 = ∅ → (suc 𝑀 ≠ ∅ → ¬ 𝐴 ≈ 𝑁)) |
10 | 1, 9 | mpi 20 | . . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≈ 𝑁) |
11 | 10 | con2i 136 | . . 3 ⊢ (𝐴 ≈ 𝑁 → ¬ 𝐴 = ∅) |
12 | neq0 4075 | . . 3 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
13 | 11, 12 | sylib 208 | . 2 ⊢ (𝐴 ≈ 𝑁 → ∃𝑥 𝑥 ∈ 𝐴) |
14 | 3 | breq2i 4792 | . . . . 5 ⊢ (𝐴 ≈ 𝑁 ↔ 𝐴 ≈ suc 𝑀) |
15 | enp1i.1 | . . . . . . . 8 ⊢ 𝑀 ∈ ω | |
16 | dif1en 8348 | . . . . . . . 8 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ≈ 𝑀) | |
17 | 15, 16 | mp3an1 1558 | . . . . . . 7 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ≈ 𝑀) |
18 | enp1i.3 | . . . . . . 7 ⊢ ((𝐴 ∖ {𝑥}) ≈ 𝑀 → 𝜑) | |
19 | 17, 18 | syl 17 | . . . . . 6 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑥 ∈ 𝐴) → 𝜑) |
20 | 19 | ex 397 | . . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → (𝑥 ∈ 𝐴 → 𝜑)) |
21 | 14, 20 | sylbi 207 | . . . 4 ⊢ (𝐴 ≈ 𝑁 → (𝑥 ∈ 𝐴 → 𝜑)) |
22 | enp1i.4 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
23 | 21, 22 | sylcom 30 | . . 3 ⊢ (𝐴 ≈ 𝑁 → (𝑥 ∈ 𝐴 → 𝜓)) |
24 | 23 | eximdv 1997 | . 2 ⊢ (𝐴 ≈ 𝑁 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥𝜓)) |
25 | 13, 24 | mpd 15 | 1 ⊢ (𝐴 ≈ 𝑁 → ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 = wceq 1630 ∃wex 1851 ∈ wcel 2144 ≠ wne 2942 ∖ cdif 3718 ∅c0 4061 {csn 4314 class class class wbr 4784 suc csuc 5868 ωcom 7211 ≈ cen 8105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-sbc 3586 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-br 4785 df-opab 4845 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-om 7212 df-1o 7712 df-er 7895 df-en 8109 df-fin 8112 |
This theorem is referenced by: en2 8351 en3 8352 en4 8353 |
Copyright terms: Public domain | W3C validator |