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Mirrors > Home > MPE Home > Th. List > enp1iOLD | Structured version Visualization version GIF version |
Description: Obsolete version of enp1i 9311 as of 6-Jan-2025. (Contributed by Mario Carneiro, 5-Jan-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
enp1iOLD.1 | ⊢ 𝑀 ∈ ω |
enp1iOLD.2 | ⊢ 𝑁 = suc 𝑀 |
enp1iOLD.3 | ⊢ ((𝐴 ∖ {𝑥}) ≈ 𝑀 → 𝜑) |
enp1iOLD.4 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
enp1iOLD | ⊢ (𝐴 ≈ 𝑁 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsuceq0 6469 | . . . . 5 ⊢ suc 𝑀 ≠ ∅ | |
2 | breq1 5151 | . . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ≈ 𝑁 ↔ ∅ ≈ 𝑁)) | |
3 | enp1iOLD.2 | . . . . . . . 8 ⊢ 𝑁 = suc 𝑀 | |
4 | ensym 9042 | . . . . . . . . 9 ⊢ (∅ ≈ 𝑁 → 𝑁 ≈ ∅) | |
5 | en0 9057 | . . . . . . . . 9 ⊢ (𝑁 ≈ ∅ ↔ 𝑁 = ∅) | |
6 | 4, 5 | sylib 218 | . . . . . . . 8 ⊢ (∅ ≈ 𝑁 → 𝑁 = ∅) |
7 | 3, 6 | eqtr3id 2789 | . . . . . . 7 ⊢ (∅ ≈ 𝑁 → suc 𝑀 = ∅) |
8 | 2, 7 | biimtrdi 253 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ≈ 𝑁 → suc 𝑀 = ∅)) |
9 | 8 | necon3ad 2951 | . . . . 5 ⊢ (𝐴 = ∅ → (suc 𝑀 ≠ ∅ → ¬ 𝐴 ≈ 𝑁)) |
10 | 1, 9 | mpi 20 | . . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≈ 𝑁) |
11 | 10 | con2i 139 | . . 3 ⊢ (𝐴 ≈ 𝑁 → ¬ 𝐴 = ∅) |
12 | neq0 4358 | . . 3 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
13 | 11, 12 | sylib 218 | . 2 ⊢ (𝐴 ≈ 𝑁 → ∃𝑥 𝑥 ∈ 𝐴) |
14 | 3 | breq2i 5156 | . . . . 5 ⊢ (𝐴 ≈ 𝑁 ↔ 𝐴 ≈ suc 𝑀) |
15 | enp1iOLD.1 | . . . . . . . 8 ⊢ 𝑀 ∈ ω | |
16 | dif1ennn 9200 | . . . . . . . 8 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ≈ 𝑀) | |
17 | 15, 16 | mp3an1 1447 | . . . . . . 7 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ≈ 𝑀) |
18 | enp1iOLD.3 | . . . . . . 7 ⊢ ((𝐴 ∖ {𝑥}) ≈ 𝑀 → 𝜑) | |
19 | 17, 18 | syl 17 | . . . . . 6 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑥 ∈ 𝐴) → 𝜑) |
20 | 19 | ex 412 | . . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → (𝑥 ∈ 𝐴 → 𝜑)) |
21 | 14, 20 | sylbi 217 | . . . 4 ⊢ (𝐴 ≈ 𝑁 → (𝑥 ∈ 𝐴 → 𝜑)) |
22 | enp1iOLD.4 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
23 | 21, 22 | sylcom 30 | . . 3 ⊢ (𝐴 ≈ 𝑁 → (𝑥 ∈ 𝐴 → 𝜓)) |
24 | 23 | eximdv 1915 | . 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 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 ∅c0 4339 {csn 4631 class class class wbr 5148 suc csuc 6388 ωcom 7887 ≈ cen 8981 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-er 8744 df-en 8985 |
This theorem is referenced by: (None) |
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