| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > enp1iOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of enp1i 9313 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 6467 | . . . . 5 ⊢ suc 𝑀 ≠ ∅ | |
| 2 | breq1 5146 | . . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ≈ 𝑁 ↔ ∅ ≈ 𝑁)) | |
| 3 | enp1iOLD.2 | . . . . . . . 8 ⊢ 𝑁 = suc 𝑀 | |
| 4 | ensym 9043 | . . . . . . . . 9 ⊢ (∅ ≈ 𝑁 → 𝑁 ≈ ∅) | |
| 5 | en0 9058 | . . . . . . . . 9 ⊢ (𝑁 ≈ ∅ ↔ 𝑁 = ∅) | |
| 6 | 4, 5 | sylib 218 | . . . . . . . 8 ⊢ (∅ ≈ 𝑁 → 𝑁 = ∅) |
| 7 | 3, 6 | eqtr3id 2791 | . . . . . . 7 ⊢ (∅ ≈ 𝑁 → suc 𝑀 = ∅) |
| 8 | 2, 7 | biimtrdi 253 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ≈ 𝑁 → suc 𝑀 = ∅)) |
| 9 | 8 | necon3ad 2953 | . . . . 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 4352 | . . 3 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
| 13 | 11, 12 | sylib 218 | . 2 ⊢ (𝐴 ≈ 𝑁 → ∃𝑥 𝑥 ∈ 𝐴) |
| 14 | 3 | breq2i 5151 | . . . . 5 ⊢ (𝐴 ≈ 𝑁 ↔ 𝐴 ≈ suc 𝑀) |
| 15 | enp1iOLD.1 | . . . . . . . 8 ⊢ 𝑀 ∈ ω | |
| 16 | dif1ennn 9201 | . . . . . . . 8 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ≈ 𝑀) | |
| 17 | 15, 16 | mp3an1 1450 | . . . . . . 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 1917 | . 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 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ∅c0 4333 {csn 4626 class class class wbr 5143 suc csuc 6386 ωcom 7887 ≈ cen 8982 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-er 8745 df-en 8986 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |