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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 8892 | . . 3 ⊢ (𝐴 ≈ suc 𝐵 → (𝐴 ∈ V ∧ suc 𝐵 ∈ V)) | |
2 | 1 | simprd 497 | . 2 ⊢ (𝐴 ≈ suc 𝐵 → suc 𝐵 ∈ V) |
3 | en0 8958 | . . . . . . 7 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
4 | 3 | biimpri 227 | . . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅) |
5 | 4 | a1i 11 | . . . . 5 ⊢ (suc 𝐵 ∈ V → (𝐴 = ∅ → 𝐴 ≈ ∅)) |
6 | nsuceq0 6401 | . . . . . 6 ⊢ suc 𝐵 ≠ ∅ | |
7 | 0sdomg 9049 | . . . . . 6 ⊢ (suc 𝐵 ∈ V → (∅ ≺ suc 𝐵 ↔ suc 𝐵 ≠ ∅)) | |
8 | 6, 7 | mpbiri 258 | . . . . 5 ⊢ (suc 𝐵 ∈ V → ∅ ≺ suc 𝐵) |
9 | 5, 8 | jctird 528 | . . . 4 ⊢ (suc 𝐵 ∈ V → (𝐴 = ∅ → (𝐴 ≈ ∅ ∧ ∅ ≺ suc 𝐵))) |
10 | ensdomtr 9058 | . . . . 5 ⊢ ((𝐴 ≈ ∅ ∧ ∅ ≺ suc 𝐵) → 𝐴 ≺ suc 𝐵) | |
11 | sdomnen 8922 | . . . . 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 2959 | . 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 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 Vcvv 3446 ∅c0 4283 class class class wbr 5106 suc csuc 6320 ≈ cen 8881 ≺ csdm 8883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-suc 6324 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 |
This theorem is referenced by: (None) |
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