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Mirrors > Home > MPE Home > Th. List > enp1i | Structured version Visualization version GIF version |
Description: Proof induction for en2i 7879 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 5722 | . . . . 5 ⊢ suc 𝑀 ≠ ∅ | |
2 | breq1 4586 | . . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ≈ 𝑁 ↔ ∅ ≈ 𝑁)) | |
3 | enp1i.2 | . . . . . . . 8 ⊢ 𝑁 = suc 𝑀 | |
4 | ensym 7891 | . . . . . . . . 9 ⊢ (∅ ≈ 𝑁 → 𝑁 ≈ ∅) | |
5 | en0 7905 | . . . . . . . . 9 ⊢ (𝑁 ≈ ∅ ↔ 𝑁 = ∅) | |
6 | 4, 5 | sylib 207 | . . . . . . . 8 ⊢ (∅ ≈ 𝑁 → 𝑁 = ∅) |
7 | 3, 6 | syl5eqr 2658 | . . . . . . 7 ⊢ (∅ ≈ 𝑁 → suc 𝑀 = ∅) |
8 | 2, 7 | syl6bi 242 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ≈ 𝑁 → suc 𝑀 = ∅)) |
9 | 8 | necon3ad 2795 | . . . . 5 ⊢ (𝐴 = ∅ → (suc 𝑀 ≠ ∅ → ¬ 𝐴 ≈ 𝑁)) |
10 | 1, 9 | mpi 20 | . . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≈ 𝑁) |
11 | 10 | con2i 133 | . . 3 ⊢ (𝐴 ≈ 𝑁 → ¬ 𝐴 = ∅) |
12 | neq0 3889 | . . 3 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
13 | 11, 12 | sylib 207 | . 2 ⊢ (𝐴 ≈ 𝑁 → ∃𝑥 𝑥 ∈ 𝐴) |
14 | 3 | breq2i 4591 | . . . . 5 ⊢ (𝐴 ≈ 𝑁 ↔ 𝐴 ≈ suc 𝑀) |
15 | enp1i.1 | . . . . . . . 8 ⊢ 𝑀 ∈ ω | |
16 | dif1en 8078 | . . . . . . . 8 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ≈ 𝑀) | |
17 | 15, 16 | mp3an1 1403 | . . . . . . 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 449 | . . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → (𝑥 ∈ 𝐴 → 𝜑)) |
21 | 14, 20 | sylbi 206 | . . . 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 1833 | . 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 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ∅c0 3874 {csn 4125 class class class wbr 4583 suc csuc 5642 ωcom 6957 ≈ cen 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-er 7629 df-en 7842 df-fin 7845 |
This theorem is referenced by: en2 8081 en3 8082 en4 8083 |
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