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Mirrors > Home > MPE Home > Th. List > enp1i | Structured version Visualization version GIF version |
Description: Proof induction for en2i 8547 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 6271 | . . . . 5 ⊢ suc 𝑀 ≠ ∅ | |
2 | breq1 5069 | . . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ≈ 𝑁 ↔ ∅ ≈ 𝑁)) | |
3 | enp1i.2 | . . . . . . . 8 ⊢ 𝑁 = suc 𝑀 | |
4 | ensym 8558 | . . . . . . . . 9 ⊢ (∅ ≈ 𝑁 → 𝑁 ≈ ∅) | |
5 | en0 8572 | . . . . . . . . 9 ⊢ (𝑁 ≈ ∅ ↔ 𝑁 = ∅) | |
6 | 4, 5 | sylib 220 | . . . . . . . 8 ⊢ (∅ ≈ 𝑁 → 𝑁 = ∅) |
7 | 3, 6 | syl5eqr 2870 | . . . . . . 7 ⊢ (∅ ≈ 𝑁 → suc 𝑀 = ∅) |
8 | 2, 7 | syl6bi 255 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ≈ 𝑁 → suc 𝑀 = ∅)) |
9 | 8 | necon3ad 3029 | . . . . 5 ⊢ (𝐴 = ∅ → (suc 𝑀 ≠ ∅ → ¬ 𝐴 ≈ 𝑁)) |
10 | 1, 9 | mpi 20 | . . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≈ 𝑁) |
11 | 10 | con2i 141 | . . 3 ⊢ (𝐴 ≈ 𝑁 → ¬ 𝐴 = ∅) |
12 | neq0 4309 | . . 3 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
13 | 11, 12 | sylib 220 | . 2 ⊢ (𝐴 ≈ 𝑁 → ∃𝑥 𝑥 ∈ 𝐴) |
14 | 3 | breq2i 5074 | . . . . 5 ⊢ (𝐴 ≈ 𝑁 ↔ 𝐴 ≈ suc 𝑀) |
15 | enp1i.1 | . . . . . . . 8 ⊢ 𝑀 ∈ ω | |
16 | dif1en 8751 | . . . . . . . 8 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ≈ 𝑀) | |
17 | 15, 16 | mp3an1 1444 | . . . . . . 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 415 | . . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → (𝑥 ∈ 𝐴 → 𝜑)) |
21 | 14, 20 | sylbi 219 | . . . 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 1918 | . 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 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 ∅c0 4291 {csn 4567 class class class wbr 5066 suc csuc 6193 ωcom 7580 ≈ cen 8506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-1o 8102 df-er 8289 df-en 8510 df-fin 8513 |
This theorem is referenced by: en2 8754 en3 8755 en4 8756 |
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