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Mirrors > Home > MPE Home > Th. List > en0ALT | Structured version Visualization version GIF version |
Description: Shorter proof of en0 8960, depending on ax-pow 5321 and ax-un 7673. (Contributed by NM, 27-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
en0ALT | ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren 8896 | . . 3 ⊢ (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅) | |
2 | f1ocnv 6797 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴) | |
3 | f1o00 6820 | . . . . . 6 ⊢ (◡𝑓:∅–1-1-onto→𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅)) | |
4 | 3 | simprbi 498 | . . . . 5 ⊢ (◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
6 | 5 | exlimiv 1934 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
7 | 1, 6 | sylbi 216 | . 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅) |
8 | 0ex 5265 | . . . 4 ⊢ ∅ ∈ V | |
9 | 8 | enref 8928 | . . 3 ⊢ ∅ ≈ ∅ |
10 | breq1 5109 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅)) | |
11 | 9, 10 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅) |
12 | 7, 11 | impbii 208 | 1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∃wex 1782 ∅c0 4283 class class class wbr 5106 ◡ccnv 5633 –1-1-onto→wf1o 6496 ≈ cen 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-12 2172 ax-ext 2704 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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 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-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-en 8887 |
This theorem is referenced by: (None) |
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