Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > iotaexeu | Structured version Visualization version GIF version |
Description: The iota class exists. This theorem does not require ax-nul 4789 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
iotaexeu | ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotaval 5862 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
2 | 1 | eqcomd 2628 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑)) |
3 | 2 | eximi 1762 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑)) |
4 | df-eu 2474 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
5 | isset 3207 | . 2 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑦 𝑦 = (℩𝑥𝜑)) | |
6 | 3, 4, 5 | 3imtr4i 281 | 1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃!weu 2470 Vcvv 3200 ℩cio 5849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-v 3202 df-sbc 3436 df-un 3579 df-sn 4178 df-pr 4180 df-uni 4437 df-iota 5851 |
This theorem is referenced by: iotasbc 38620 pm14.18 38629 iotavalb 38631 sbiota1 38635 |
Copyright terms: Public domain | W3C validator |