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Mirrors > Home > MPE Home > Th. List > Mathboxes > mof0ALT | Structured version Visualization version GIF version |
Description: Alternate proof for mof0 45781 with stronger requirements on distinct variables. Uses mo4 2565. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mof0ALT | ⊢ ∃*𝑓 𝑓:𝐴⟶∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f00 6579 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅)) | |
2 | 1 | simplbi 501 | . . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝑓 = ∅) |
3 | f00 6579 | . . . . 5 ⊢ (𝑔:𝐴⟶∅ ↔ (𝑔 = ∅ ∧ 𝐴 = ∅)) | |
4 | 3 | simplbi 501 | . . . 4 ⊢ (𝑔:𝐴⟶∅ → 𝑔 = ∅) |
5 | eqtr3 2758 | . . . 4 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → 𝑓 = 𝑔) | |
6 | 2, 4, 5 | syl2an 599 | . . 3 ⊢ ((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔) |
7 | 6 | gen2 1804 | . 2 ⊢ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔) |
8 | feq1 6504 | . . 3 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶∅ ↔ 𝑔:𝐴⟶∅)) | |
9 | 8 | mo4 2565 | . 2 ⊢ (∃*𝑓 𝑓:𝐴⟶∅ ↔ ∀𝑓∀𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)) |
10 | 7, 9 | mpbir 234 | 1 ⊢ ∃*𝑓 𝑓:𝐴⟶∅ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1541 = wceq 1543 ∃*wmo 2537 ∅c0 4223 ⟶wf 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-fun 6360 df-fn 6361 df-f 6362 |
This theorem is referenced by: (None) |
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