![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > enp1i | Structured version Visualization version GIF version |
Description: Proof induction for en2 9221 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.) Generalize to all ordinals and avoid ax-pow 5318, ax-un 7668. (Revised by BTernaryTau, 6-Jan-2025.) |
Ref | Expression |
---|---|
enp1i.1 | ⊢ Ord 𝑀 |
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 | enp1i.2 | . . 3 ⊢ 𝑁 = suc 𝑀 | |
2 | 1 | breq2i 5111 | . 2 ⊢ (𝐴 ≈ 𝑁 ↔ 𝐴 ≈ suc 𝑀) |
3 | enp1i.1 | . . . . 5 ⊢ Ord 𝑀 | |
4 | encv 8887 | . . . . . . 7 ⊢ (𝐴 ≈ suc 𝑀 → (𝐴 ∈ V ∧ suc 𝑀 ∈ V)) | |
5 | 4 | simprd 496 | . . . . . 6 ⊢ (𝐴 ≈ suc 𝑀 → suc 𝑀 ∈ V) |
6 | sssucid 6395 | . . . . . . 7 ⊢ 𝑀 ⊆ suc 𝑀 | |
7 | ssexg 5278 | . . . . . . 7 ⊢ ((𝑀 ⊆ suc 𝑀 ∧ suc 𝑀 ∈ V) → 𝑀 ∈ V) | |
8 | 6, 7 | mpan 688 | . . . . . 6 ⊢ (suc 𝑀 ∈ V → 𝑀 ∈ V) |
9 | elong 6323 | . . . . . 6 ⊢ (𝑀 ∈ V → (𝑀 ∈ On ↔ Ord 𝑀)) | |
10 | 5, 8, 9 | 3syl 18 | . . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → (𝑀 ∈ On ↔ Ord 𝑀)) |
11 | 3, 10 | mpbiri 257 | . . . 4 ⊢ (𝐴 ≈ suc 𝑀 → 𝑀 ∈ On) |
12 | rexdif1en 9098 | . . . 4 ⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) | |
13 | 11, 12 | mpancom 686 | . . 3 ⊢ (𝐴 ≈ suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) |
14 | enp1i.3 | . . . 4 ⊢ ((𝐴 ∖ {𝑥}) ≈ 𝑀 → 𝜑) | |
15 | 14 | reximi 3085 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀 → ∃𝑥 ∈ 𝐴 𝜑) |
16 | df-rex 3072 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
17 | enp1i.4 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
18 | 17 | imp 407 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) |
19 | 18 | eximi 1837 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥𝜓) |
20 | 16, 19 | sylbi 216 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜓) |
21 | 13, 15, 20 | 3syl 18 | . 2 ⊢ (𝐴 ≈ suc 𝑀 → ∃𝑥𝜓) |
22 | 2, 21 | sylbi 216 | 1 ⊢ (𝐴 ≈ 𝑁 → ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∃wrex 3071 Vcvv 3443 ∖ cdif 3905 ⊆ wss 3908 {csn 4584 class class class wbr 5103 Ord word 6314 Oncon0 6315 suc csuc 6317 ≈ cen 8876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-en 8880 |
This theorem is referenced by: en2 9221 en3 9222 en4 9223 |
Copyright terms: Public domain | W3C validator |