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Mirrors > Home > MPE Home > Th. List > Mathboxes > ensucne0OLD | Structured version Visualization version GIF version |
Description: A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ensucne0OLD | ⊢ (𝐴 ≈ suc 𝐵 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | encv 8517 | . . 3 ⊢ (𝐴 ≈ suc 𝐵 → (𝐴 ∈ V ∧ suc 𝐵 ∈ V)) | |
2 | 1 | simprd 498 | . 2 ⊢ (𝐴 ≈ suc 𝐵 → suc 𝐵 ∈ V) |
3 | en0 8572 | . . . . . . 7 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
4 | 3 | biimpri 230 | . . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅) |
5 | 4 | a1i 11 | . . . . 5 ⊢ (suc 𝐵 ∈ V → (𝐴 = ∅ → 𝐴 ≈ ∅)) |
6 | nsuceq0 6271 | . . . . . 6 ⊢ suc 𝐵 ≠ ∅ | |
7 | 0sdomg 8646 | . . . . . 6 ⊢ (suc 𝐵 ∈ V → (∅ ≺ suc 𝐵 ↔ suc 𝐵 ≠ ∅)) | |
8 | 6, 7 | mpbiri 260 | . . . . 5 ⊢ (suc 𝐵 ∈ V → ∅ ≺ suc 𝐵) |
9 | 5, 8 | jctird 529 | . . . 4 ⊢ (suc 𝐵 ∈ V → (𝐴 = ∅ → (𝐴 ≈ ∅ ∧ ∅ ≺ suc 𝐵))) |
10 | ensdomtr 8653 | . . . . 5 ⊢ ((𝐴 ≈ ∅ ∧ ∅ ≺ suc 𝐵) → 𝐴 ≺ suc 𝐵) | |
11 | sdomnen 8538 | . . . . 5 ⊢ (𝐴 ≺ suc 𝐵 → ¬ 𝐴 ≈ suc 𝐵) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ ((𝐴 ≈ ∅ ∧ ∅ ≺ suc 𝐵) → ¬ 𝐴 ≈ suc 𝐵) |
13 | 9, 12 | syl6 35 | . . 3 ⊢ (suc 𝐵 ∈ V → (𝐴 = ∅ → ¬ 𝐴 ≈ suc 𝐵)) |
14 | 13 | necon2ad 3031 | . 2 ⊢ (suc 𝐵 ∈ V → (𝐴 ≈ suc 𝐵 → 𝐴 ≠ ∅)) |
15 | 2, 14 | mpcom 38 | 1 ⊢ (𝐴 ≈ suc 𝐵 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∅c0 4291 class class class wbr 5066 suc csuc 6193 ≈ cen 8506 ≺ csdm 8508 |
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-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-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 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-suc 6197 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 |
This theorem is referenced by: (None) |
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