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Mirrors > Home > MPE Home > Th. List > Mathboxes > ensucne0 | Structured version Visualization version GIF version |
Description: A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof shortened by SN, 16-Nov-2023.) |
Ref | Expression |
---|---|
ensucne0 | ⊢ (𝐴 ≈ suc 𝐵 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsuceq0 6401 | . . . 4 ⊢ suc 𝐵 ≠ ∅ | |
2 | en0r 8961 | . . . 4 ⊢ (∅ ≈ suc 𝐵 ↔ suc 𝐵 = ∅) | |
3 | 1, 2 | nemtbir 3041 | . . 3 ⊢ ¬ ∅ ≈ suc 𝐵 |
4 | breq1 5109 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ suc 𝐵 ↔ ∅ ≈ suc 𝐵)) | |
5 | 3, 4 | mtbiri 327 | . 2 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≈ suc 𝐵) |
6 | 5 | necon2ai 2974 | 1 ⊢ (𝐴 ≈ suc 𝐵 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ≠ wne 2944 ∅c0 4283 class class class wbr 5106 suc csuc 6320 ≈ cen 8881 |
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-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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-sb 2069 df-mo 2539 df-clab 2715 df-cleq 2729 df-clel 2815 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-sn 4588 df-pr 4590 df-op 4594 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-suc 6324 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-en 8885 |
This theorem is referenced by: (None) |
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